Introduce

This is a toy example of application of Bayesian Optimization. We will use the Bayesian Optimization to find the maximum value of one simple function.

### install.packages("GPfit")
library(GPfit)
library(dplyr)
library(ggplot2)
library(purrr)

Preparation

We will optimize this simple function:

\[\left(6x-2\right)^2sin\left(12x-4\right)\]

First we define the function in R, then draw the graph of this function in the axes, and generate n0 initial points.

### Generate the function
f <- function(x) {
  return((6*x-2)^2*sin(12*x-4))
}

### Draw the graph of this function
x <- seq(0, 1, by=0.0001)
y <- f(x)
graph_data <- data.frame(x, y)
graph_data %>% ggplot() +
  geom_point(mapping = aes(x = x, y = y), size = 0.01, color = "navyblue") +
  theme_bw()

y_min <- min(y)

min <- 0
for (i in x) {
  if(f(i) < min) {
    min <- f(i)
    x_min <- i
  }
}

### Generate the initial points
evaluation <- data.frame("x" = c(0, 1/3, 2/3, 1), "y" = f(c(0, 1/3, 2/3, 1)))

Gaussian Process Model

Here we build the initial GP model. We choose the power exponential correlation function here.

fit <- GP_fit(
  X = evaluation[, "x"],
  Y = evaluation[, "y"],
  corr = list(type = "exponential", power = 1.95)
)



x_new <- seq(0, 1, length.out = 100)
pred <- predict.GP(fit, xnew = data.frame(x = x_new))
mu <- pred$Y_hat
sigma <- sqrt(pred$MSE)

pred <- pred %>% data.frame()
pred %>% ggplot() +
  geom_line(mapping = aes(x = complete_data.xnew.1, y = Y_hat), color = "gold") +
  geom_ribbon(mapping = aes(x = complete_data.xnew.1, y = Y_hat, xmin = 0, xmax = 1, ymin = Y_hat-sqrt(MSE), ymax = Y_hat+sqrt(MSE)), fill = "navyblue", alpha = 0.3) +
  geom_point(mapping = aes(x = evaluation[1, "x"], y = evaluation[1, "y"]), color = "navyblue") +
  geom_point(mapping = aes(x = evaluation[2, "x"], y = evaluation[2, "y"]), color = "navyblue") +
  geom_point(mapping = aes(x = evaluation[3, "x"], y = evaluation[3, "y"]), color = "navyblue") +
  geom_point(mapping = aes(x = evaluation[4, "x"], y = evaluation[4, "y"]), color = "navyblue") +
  xlab("x") +
  ylab("f(x)") +
  theme_bw()

Then we will use diferent acquisition functions to find the x to evaluate next.

### y_best is the current best y value
y_best <- min(evaluation[, "y"])

### Make the f(x) - x plot a function

plot <- function(x_next) {
  pred %>% ggplot() +
    geom_line(mapping = aes(x = complete_data.xnew.1, y = Y_hat), color = "gold") +
    geom_ribbon(mapping = aes(x = complete_data.xnew.1, y = Y_hat, xmin = 0, xmax = 1, ymin = Y_hat-sqrt(MSE), ymax = Y_hat+sqrt(MSE)), fill = "navyblue", alpha = 0.3) +
    geom_point(mapping = aes(x = evaluation[1, "x"], y = evaluation[1, "y"]), color = "navyblue") +
    geom_point(mapping = aes(x = evaluation[2, "x"], y = evaluation[2, "y"]), color = "navyblue") +
    geom_point(mapping = aes(x = evaluation[3, "x"], y = evaluation[3, "y"]), color = "navyblue") +
    geom_point(mapping = aes(x = evaluation[4, "x"], y = evaluation[4, "y"]), color = "navyblue") +
    geom_vline(mapping = aes(xintercept = x_next), color = "red", linetype = "dashed") +
    geom_point(mapping = aes(x = x_next, y = Y_hat[ceiling(x_next/0.01)]), color = "red") +
    xlab("x") +
    ylab("f(x)") +
    theme_bw()
}

1.Probability of improvement

probability_improvement <- map2_dbl(
  mu,
  sigma,
  function(m, s) {
    if (s == 0) return(0)
    else {
      poi <- pnorm((y_best - m) / s)
      # poi <- 1 - poi (if maximizing)
      return(poi)
    }
  }
)

x_next_POI <- x_new[which.max(probability_improvement)]

ggplot() +
  geom_line(mapping = aes(x = x_new, y = probability_improvement), color = "navyblue") +
  geom_vline(mapping = aes(xintercept = x_next_POI), color = "red", linetype = "dashed") +
  xlab("x") +
  theme_bw()

plot(x_next_POI)

  1. Expected improvement
expected_improvement <- map2_dbl(
  mu, sigma,
  function(m, s) {
    if (s == 0) return(0)
    gamma <- (y_best - m) / s
    phi <- pnorm(gamma)
    return(s * (gamma * phi + dnorm(gamma)))
  }
)

x_next_EI <- x_new[which.max(expected_improvement)]

ggplot() +
  geom_line(mapping = aes(x = x_new, y = expected_improvement), color = "navyblue") +
  geom_vline(mapping = aes(xintercept = x_next_EI), color = "red", linetype = "dashed") +
  xlab("x") +
  theme_bw() 

plot(x_next_EI)

  1. GP lower confidence bound
kappa <- 2 # tunable
lower_confidence_bound <- mu - kappa * sigma
# if maximizing: upper_confidence_bound <- mu + kappa * sigma

### Notice here is min, not max.
x_next_LCB <- x_new[which.min(lower_confidence_bound)]

ggplot() +
  geom_line(mapping = aes(x = x_new, y = lower_confidence_bound), color = "navyblue") +
  geom_vline(mapping = aes(xintercept = x_next_LCB), color = "red", linetype = "dashed") +
  xlab("x") +
  theme_bw() 

plot(x_next_LCB)

Optimization

In the last section, we discussed how Bayesian optimization works in one loop. In this section, we will continue looping until we got the optimized value of the simple function.

Firstly we will choose a “good” acquisition function.

Probability of improvment

This acquisition function measures the likelihood that the next x is better(lower or higher depends on our goal) than the current optimal value.

evaluation_t <- evaluation
t <- 1
data <- data.frame()
data$Y_hat <- c()
data$MSE <- c()
data$iteration <- c()
data$x_x <- c()
data$y_y <- c()
pred <- predict.GP(fit, xnew = data.frame(x = x_new))
pred <- pred %>% data.frame() %>% select(c("Y_hat", "MSE", "complete_data.xnew.1"))
pred <- pred %>% mutate("iteration" = 0) %>% mutate("x_x" = NA) %>% mutate("y_y" = NA)
data <- rbind(data, pred)
acq_data = data.frame()
probability_improvement <- probability_improvement %>% 
  as.data.frame() %>%
  mutate("iteration" = 1) %>% 
  mutate("x_next_POI" = NA)
colnames(probability_improvement) <- c("probability_improvement", "iteration", "x_next_POI")
acq_data <- rbind(acq_data, probability_improvement)
x_next_df_POI <- data.frame(x_next_POI) %>% mutate("probability_improvement" = NA) %>% mutate("iteration" = 1)
colnames(x_next_df_POI) <- c("x_next_POI", "probability_improvement", "iteration")
acq_data <- rbind(acq_data, x_next_df_POI)

while(t <= 10) {
  evaluation[nrow(evaluation)+1,] = c(x_next_POI, f(x_next_POI))

  fit <- GP_fit(
    X = evaluation[, "x"],
    Y = evaluation[, "y"],
    corr = list(type = "exponential", power = 1.95)
  )

  x_new <- seq(0, 1, length.out = 100)
  pred <- predict.GP(fit, xnew = data.frame(x = x_new))
  mu <- pred$Y_hat
  sigma <- sqrt(pred$MSE)
  pred <- pred %>% data.frame() %>% select(c("Y_hat", "MSE", "complete_data.xnew.1"))
  pred <- pred %>% mutate("iteration" = t) %>% mutate("x_x" = NA) %>% mutate("y_y" = NA)
  data <- rbind(data, pred)
  
  probability_improvement <- map2_dbl(
    mu,
    sigma,
    function(m, s) {
      if (s == 0) return(0)
      else {
        poi <- pnorm((y_best - m) / s)
        # poi <- 1 - poi (if maximizing)
        return(poi)
      }
    }
  )
  x_next_POI <- x_new[which.max(probability_improvement)]
  probability_improvement <- probability_improvement %>% as.data.frame()
  colnames(probability_improvement) <- "probability_improvement"
  probability_improvement <- probability_improvement %>% 
    mutate("iteration" = t+1) %>% 
    mutate("x_next_POI" = NA)
  acq_data <- rbind(acq_data, probability_improvement)
  x_next_df <- data.frame(x_next_POI) %>% mutate("probability_improvement" = NA) %>% mutate("iteration" = t+1)
  acq_data <- rbind(acq_data, x_next_df)
  t <- t+1
}
t <- 1
nr <- nrow(data)
for (i in 4:14) {
  for (j in 1:i) {
    data[nr + t,] = c(NA, NA, NA, i-4, evaluation[j, 1], evaluation[j, 2])
    t <- t+1
  }
}

plot_1 <- data %>% ggplot() +
    geom_line(mapping = aes(x = complete_data.xnew.1, y = Y_hat), color = "gold") +
    geom_ribbon(mapping = aes(x = complete_data.xnew.1, y = Y_hat, xmin = 0, xmax = 1, ymin = Y_hat-sqrt(MSE), ymax = Y_hat+sqrt(MSE)), fill = "navyblue", alpha = 0.3) +
    geom_point(mapping = aes(x = x_x, y = y_y)) +
    facet_wrap(~iteration) +
    xlab("x") +
    ylab("f(x)") +
    theme_bw()

x_new_df <- rep(c(x_new, NA),10) %>% as.data.frame()
colnames(x_new_df) <- "x_new"
acq_data <- cbind(acq_data %>% filter(iteration<11), x_new_df)
plot_2 <- acq_data %>% ggplot() +
  geom_line(mapping = aes(x = x_new, y = probability_improvement), color = "navyblue") +
  geom_vline(mapping = aes(xintercept = x_next_POI), color = "red", linetype = "dashed") +
  xlab("x") +
  facet_wrap(~ iteration) +
  theme_bw()

print(plot_1)

print(plot_2)

Expected improvment

This acquisition function takes how large the improvement is into account.

evaluation <- evaluation_t
t <- 1
data <- data.frame()
data$Y_hat <- c()
data$MSE <- c()
data$iteration <- c()
data$x_x <- c()
data$y_y <- c()
pred <- predict.GP(fit, xnew = data.frame(x = x_new))
pred <- pred %>% data.frame() %>% select(c("Y_hat", "MSE", "complete_data.xnew.1"))
pred <- pred %>% mutate("iteration" = 0) %>% mutate("x_x" = NA) %>% mutate("y_y" = NA)
data <- rbind(data, pred)
acq_data = data.frame()
expected_improvement <- expected_improvement %>% 
  as.data.frame() %>%
  mutate("iteration" = 1) %>% 
  mutate("x_next_EI" = NA)
colnames(expected_improvement) <- c("expected_improvement", "iteration", "x_next_EI")
acq_data <- rbind(acq_data, expected_improvement)
x_next_df_EI <- data.frame(x_next_EI) %>% mutate("expected_improvement" = NA) %>% mutate("iteration" = 1)
colnames(x_next_df_EI) <- c("x_next_EI", "expected_improvement", "iteration")
acq_data <- rbind(acq_data, x_next_df_EI)

while(t <= 10) {
  evaluation[nrow(evaluation)+1,] = c(x_next_EI, f(x_next_EI))

  fit <- GP_fit(
    X = evaluation[, "x"],
    Y = evaluation[, "y"],
    corr = list(type = "exponential", power = 1.95)
  )

  x_new <- seq(0, 1, length.out = 100)
  pred <- predict.GP(fit, xnew = data.frame(x = x_new))
  mu <- pred$Y_hat
  sigma <- sqrt(pred$MSE)
  pred <- pred %>% data.frame() %>% select(c("Y_hat", "MSE", "complete_data.xnew.1"))
  pred <- pred %>% mutate("iteration" = t) %>% mutate("x_x" = NA) %>% mutate("y_y" = NA)
  data <- rbind(data, pred)
  
  expected_improvement <- map2_dbl(
    mu, sigma,
    function(m, s) {
      if (s == 0) return(0)
      gamma <- (y_best - m) / s
      phi <- pnorm(gamma)
      return(s * (gamma * phi + dnorm(gamma)))
    }
  )
  
  x_next_EI <- x_new[which.max(expected_improvement)]
  expected_improvement <- expected_improvement %>% as.data.frame()
  colnames(expected_improvement) <- "expected_improvement"
  expected_improvement <- expected_improvement %>% 
    mutate("iteration" = t+1) %>% 
    mutate("x_next_EI" = NA)
  acq_data <- rbind(acq_data, expected_improvement)
  x_next_df <- data.frame(x_next_EI) %>% mutate("expected_improvement" = NA) %>% mutate("iteration" = t+1)
  acq_data <- rbind(acq_data, x_next_df)
  t <- t+1
}
t <- 1
nr <- nrow(data)
for (i in 4:14) {
  for (j in 1:i) {
    data[nr + t,] = c(NA, NA, NA, i-4, evaluation[j, 1], evaluation[j, 2])
    t <- t+1
  }
}

plot_1 <- data %>% ggplot() +
    geom_line(mapping = aes(x = complete_data.xnew.1, y = Y_hat), color = "gold") +
    geom_ribbon(mapping = aes(x = complete_data.xnew.1, y = Y_hat, xmin = 0, xmax = 1, ymin = Y_hat-sqrt(MSE), ymax = Y_hat+sqrt(MSE)), fill = "navyblue", alpha = 0.3) +
    geom_point(mapping = aes(x = x_x, y = y_y)) +
    facet_wrap(~iteration) +
    xlab("x") +
    ylab("f(x)") +
    theme_bw()

x_new_df <- rep(c(x_new, NA),10) %>% as.data.frame()
colnames(x_new_df) <- "x_new"
acq_data <- cbind(acq_data %>% filter(iteration < 11), x_new_df)
plot_2 <- acq_data %>% ggplot() +
  geom_line(mapping = aes(x = x_new, y = expected_improvement), color = "navyblue") +
  geom_vline(mapping = aes(xintercept = x_next_EI), color = "red", linetype = "dashed") +
  xlab("x") +
  facet_wrap(~ iteration) +
  theme_bw()

print(plot_1)

print(plot_2)

### Here we save the data to use it later.
data_EI <- data

GP lower confidence bound

This acquisition functions balances the area where mean(x) is large and the area where sigma(x) is large. In other words, this acquisition function trades off between exploration and exploitation.

evaluation <- evaluation_t
t <- 1
data <- data.frame()
data$Y_hat <- c()
data$MSE <- c()
data$iteration <- c()
data$x_x <- c()
data$y_y <- c()
pred <- predict.GP(fit, xnew = data.frame(x = x_new))
pred <- pred %>% data.frame() %>% select(c("Y_hat", "MSE", "complete_data.xnew.1"))
pred <- pred %>% mutate("iteration" = 0) %>% mutate("x_x" = NA) %>% mutate("y_y" = NA)
data <- rbind(data, pred)
  
acq_data = data.frame()
lower_confidence_bound <- lower_confidence_bound %>% 
  as.data.frame() %>%
  mutate("iteration" = 1) %>% 
  mutate("x_next_LCB" = NA)
colnames(lower_confidence_bound) <- c("lower_confidence_bound", "iteration", "x_next_LCB")
acq_data <- rbind(acq_data, lower_confidence_bound)
x_next_df_LCB <- data.frame(x_next_LCB) %>% mutate("lower_confidence_bound" = NA) %>% mutate("iteration" = 1)
colnames(x_next_df_LCB) <- c("x_next_LCB", "lower_confidence_bound", "iteration")
acq_data <- rbind(acq_data, x_next_df_LCB)

while(t <= 8) {
  evaluation[nrow(evaluation)+1,] = c(x_next_LCB, f(x_next_LCB))

  fit <- GP_fit(
    X = evaluation[, "x"],
    Y = evaluation[, "y"],
    corr = list(type = "exponential", power = 1.95)
  )

  x_new <- seq(0, 1, length.out = 100)
  pred <- predict.GP(fit, xnew = data.frame(x = x_new))
  mu <- pred$Y_hat
  sigma <- sqrt(pred$MSE)
  pred <- pred %>% data.frame() %>% select(c("Y_hat", "MSE", "complete_data.xnew.1"))
  pred <- pred %>% mutate("iteration" = t) %>% mutate("x_x" = NA) %>% mutate("y_y" = NA)
  data <- rbind(data, pred)
  
  lower_confidence_bound <- mu - kappa * sigma
  
  x_next_LCB <- x_new[which.min(lower_confidence_bound)]
  lower_confidence_bound <- lower_confidence_bound %>% as.data.frame()
  colnames(lower_confidence_bound) <- "lower_confidence_bound"
  lower_confidence_bound <- lower_confidence_bound %>% 
    mutate("iteration" = t+1) %>% 
    mutate("x_next_LCB" = NA)
  acq_data <- rbind(acq_data, lower_confidence_bound)
  x_next_df <- data.frame(x_next_LCB) %>% mutate("lower_confidence_bound" = NA) %>% mutate("iteration" = t+1)
  acq_data <- rbind(acq_data, x_next_df)
  t <- t+1
}
t <- 1
nr <- nrow(data)
for (i in 4:12) {
  for (j in 1:i) {
    data[nr + t,] = c(NA, NA, NA, i-4, evaluation[j, 1], evaluation[j, 2])
    t <- t+1
  }
}

plot_1 <- data %>% ggplot() +
    geom_line(mapping = aes(x = complete_data.xnew.1, y = Y_hat), color = "gold") +
    geom_ribbon(mapping = aes(x = complete_data.xnew.1, y = Y_hat, xmin = 0, xmax = 1, ymin = Y_hat-sqrt(MSE), ymax = Y_hat+sqrt(MSE)), fill = "navyblue", alpha = 0.3) +
    geom_point(mapping = aes(x = x_x, y = y_y)) +
    facet_wrap(~iteration) +
    xlab("x") +
    ylab("f(x)") +
    theme_bw()

x_new_df <- rep(c(x_new, NA),8) %>% as.data.frame()
colnames(x_new_df) <- "x_new"
acq_data <- cbind(acq_data %>% filter(iteration < 9), x_new_df)
plot_2 <- acq_data %>% ggplot() +
  geom_line(mapping = aes(x = x_new, y = lower_confidence_bound), color = "navyblue") +
  geom_vline(mapping = aes(xintercept = x_next_LCB), color = "red", linetype = "dashed") +
  xlab("x") +
  facet_wrap(~ iteration) +
  theme_bw()

print(plot_1)

print(plot_2)

Here we give a detailed example to show the Expected Improvement acqusition function.

t <- 1
while(t < 10) {
  print(data_EI %>% filter(iteration == t) %>% ggplot() +
    geom_line(mapping = aes(x = complete_data.xnew.1, y = Y_hat), color = "gold") +
    geom_ribbon(mapping = aes(x = complete_data.xnew.1, y = Y_hat, xmin = 0, xmax = 1, ymin = Y_hat-sqrt(MSE), ymax = Y_hat+sqrt(MSE)), fill = "navyblue", alpha = 0.3) +
    geom_point(mapping = aes(x = x_x, y = y_y)) +
    xlab("x") +
    ylab("f(x)") +
    labs(paste("The", t, "iteration")) +
    theme_bw())
  t <- t+1
}

Then we pick the x value that has the minimal f(x) value from what we evaluated.

### Pick the x that has the minimal f(x) value from what we evaluated.
cat("The opmized x we find is", evaluation[which.min(evaluation$y),]$x)
## The opmized x we find is 0.7575758
### Compare it with the real x that minimize f(x) in [0,1]
cat("The real optimized x is", x_min)
## The real optimized x is 0.7572

We can see that we use 10 evaluations to get the x, and the result is pretty close to the real one.

Comparison

In this section, we will compare the Bayesian optimization with quasi-Newton method. We will try several initial guesses.

### Firstly, we build several equally spaced initial guesses in [0, 1].
ini_guess <- seq(0, 1, length.out = 9)

### Then we will use quasi-Newton method to try to find the optimal f(x) value.
n=1
for(t in ini_guess) {
  res <- optim(t, f, method = "BFGS", lower = 0, upper = 1)
  writeLines(paste("The", n, "initial guess x0 = ", t, ":\nx=", res$par,"\nf(x)=", res$value, "\nTimes of iteration=", res$counts[1], "\n-----------------------------------------------"))
  n <- n+1
}
## The 1 initial guess x0 =  0 :
## x= 0.142589801857248 
## f(x)= -0.986325406263923 
## Times of iteration= 49 
## -----------------------------------------------
## The 2 initial guess x0 =  0.125 :
## x= 0.142591747065892 
## f(x)= -0.986325405324459 
## Times of iteration= 19 
## -----------------------------------------------
## The 3 initial guess x0 =  0.25 :
## x= 0.142588440215961 
## f(x)= -0.986325406235592 
## Times of iteration= 10 
## -----------------------------------------------
## The 4 initial guess x0 =  0.375 :
## x= 0.142591304993005 
## f(x)= -0.986325405639193 
## Times of iteration= 19 
## -----------------------------------------------
## The 5 initial guess x0 =  0.5 :
## x= 0.142589561542382 
## f(x)= -0.986325406299975 
## Times of iteration= 47 
## -----------------------------------------------
## The 6 initial guess x0 =  0.625 :
## x= 0.757247331284393 
## f(x)= -6.02074005468035 
## Times of iteration= 9 
## -----------------------------------------------
## The 7 initial guess x0 =  0.75 :
## x= 0.757247332200751 
## f(x)= -6.02074005468175 
## Times of iteration= 6 
## -----------------------------------------------
## The 8 initial guess x0 =  0.875 :
## x= 0.757247564706613 
## f(x)= -6.02074005500689 
## Times of iteration= 26 
## -----------------------------------------------
## The 9 initial guess x0 =  1 :
## x= 0.75724733297612 
## f(x)= -6.02074005468293 
## Times of iteration= 10 
## -----------------------------------------------

We can see there are two disadvantages of quasi-Newton method. The first one is we have to choose a “good” initial guess to find the global optimum. In our example, if the initial guess is bigger than 0.5, the quasi-Newton method will find a local optimum.

The second one is it will need more times of iteration to get the optimal value. The minimum number of iteration in our example is 13 times, which is 3 times more than our Bayesian optimization example.

Also, because here we use a simple function. If we want to optimize a complex function, which cannot or very difficult to get the derivatives, the quasi-Newton method cannot or hard to find the optimum, but the Bayesian optimization can do that.

Initial Settings

In the previous part, we use 4 evenly spaced initial points. In this part, we will try to use different sets of initial points.

Three evenly spaced points.

Firstly, we will try 3 evenly spaced points between 0 and 1 as the initial set of points.

evaluation <- data.frame("x" = c(0, 1/2, 1), "y" = f(c(0, 1/2, 1)))

pip <- function(evaluation, num) {
  fit <- GP_fit(
  X = evaluation[, "x"],
  Y = evaluation[, "y"],
  corr = list(type = "exponential", power = 1.95)
)

t <- 1
data <- data.frame()
data$Y_hat <- c()
data$MSE <- c()
data$iteration <- c()
data$x_x <- c()
data$y_y <- c()
pred <- predict.GP(fit, xnew = data.frame(x = x_new))
pred <- pred %>% data.frame() %>% select(c("Y_hat", "MSE", "complete_data.xnew.1"))
pred <- pred %>% mutate("iteration" = 0) %>% mutate("x_x" = NA) %>% mutate("y_y" = NA)
data <- rbind(data, pred)

acq_data = data.frame()
expected_improvement <- expected_improvement %>% 
  as.data.frame() %>%
  mutate("iteration" = 1) %>% 
  mutate("x_next_EI" = NA)
colnames(expected_improvement) <- c("expected_improvement", "iteration", "x_next_EI")
acq_data <- rbind(acq_data, expected_improvement)
x_next_df_EI <- data.frame(x_next_EI) %>% mutate("expected_improvement" = NA) %>% mutate("iteration" = 1)
colnames(x_next_df_EI) <- c("x_next_EI", "expected_improvement", "iteration")
acq_data <- rbind(acq_data, x_next_df_EI)

while(t <= 10) {
  evaluation[nrow(evaluation)+1,] = c(x_next_EI, f(x_next_EI))

  fit <- GP_fit(
    X = evaluation[, "x"],
    Y = evaluation[, "y"],
    corr = list(type = "exponential", power = 1.95)
  )

  x_new <- seq(0, 1, length.out = 100)
  pred <- predict.GP(fit, xnew = data.frame(x = x_new))
  mu <- pred$Y_hat
  sigma <- sqrt(pred$MSE)
  pred <- pred %>% data.frame() %>% select(c("Y_hat", "MSE", "complete_data.xnew.1"))
  pred <- pred %>% mutate("iteration" = t) %>% mutate("x_x" = NA) %>% mutate("y_y" = NA)
  data <- rbind(data, pred)
  
  expected_improvement <- map2_dbl(
    mu, sigma,
    function(m, s) {
      if (s == 0) return(0)
      gamma <- (y_best - m) / s
      phi <- pnorm(gamma)
      return(s * (gamma * phi + dnorm(gamma)))
    }
  )
  
  x_next_EI <- x_new[which.max(expected_improvement)]
  expected_improvement <- expected_improvement %>% as.data.frame()
  colnames(expected_improvement) <- "expected_improvement"
  expected_improvement <- expected_improvement %>% 
    mutate("iteration" = t+1) %>% 
    mutate("x_next_EI" = NA)
  acq_data <- rbind(acq_data, expected_improvement)
  x_next_df <- data.frame(x_next_EI) %>% mutate("expected_improvement" = NA) %>% mutate("iteration" = t+1)
  acq_data <- rbind(acq_data, x_next_df)
  t <- t+1
}
t <- 1
nr <- nrow(data)
for (i in (num):(num+10)) {
  for (j in 1:i) {
    data[nr + t,] = c(NA, NA, NA, i-num, evaluation[j, 1], evaluation[j, 2])
    t <- t+1
  }
}

plot_1 <- data %>% ggplot() +
    geom_line(mapping = aes(x = complete_data.xnew.1, y = Y_hat), color = "gold") +
    geom_ribbon(mapping = aes(x = complete_data.xnew.1, y = Y_hat, xmin = 0, xmax = 1, ymin = Y_hat-sqrt(MSE), ymax = Y_hat+sqrt(MSE)), fill = "navyblue", alpha = 0.3) +
    geom_point(mapping = aes(x = x_x, y = y_y)) +
    facet_wrap(~iteration) +
    xlab("x") +
    ylab("f(x)") +
    theme_bw()

x_new_df <- rep(c(x_new, NA),10) %>% as.data.frame()
colnames(x_new_df) <- "x_new"
acq_data <- cbind(acq_data %>% filter(iteration < 11), x_new_df)
plot_2 <- acq_data %>% ggplot() +
  geom_line(mapping = aes(x = x_new, y = expected_improvement), color = "navyblue") +
  geom_vline(mapping = aes(xintercept = x_next_EI), color = "red", linetype = "dashed") +
  xlab("x") +
  facet_wrap(~ iteration) +
  theme_bw()

print(plot_1)
print(plot_2)
return(data)
} 

data <- pip(evaluation, 3)

Four randomly located initial points.

Here we use four randomly located points instead of four evenly spaced points as the initial points set.

set.seed(1234)
x <- sample.int(100,4)/100
evaluation <- data.frame("x" = x, "y" = f(x))
pip(evaluation, 4)

##             Y_hat          MSE complete_data.xnew.1 iteration       x_x
## 1    -0.314345885 4.486900e-01           0.00000000         0        NA
## 2    -0.321626382 3.411207e-01           0.01010101         0        NA
## 3    -0.332389212 2.504301e-01           0.02020202         0        NA
## 4    -0.346264928 1.758942e-01           0.03030303         0        NA
## 5    -0.362836383 1.165089e-01           0.04040404         0        NA
## 6    -0.381650842 7.103962e-02           0.05050505         0        NA
## 7    -0.402237945 3.808478e-02           0.06060606         0        NA
## 8    -0.424139380 1.615572e-02           0.07070707         0        NA
## 9    -0.446971415 3.800753e-03           0.08080808         0        NA
## 10   -0.470699024 4.875145e-05           0.09090909         0        NA
## 11   -0.495599231 4.080030e-03           0.10101010         0        NA
## 12   -0.519572139 1.113635e-02           0.11111111         0        NA
## 13   -0.541212854 1.848639e-02           0.12121212         0        NA
## 14   -0.559413819 2.446129e-02           0.13131313         0        NA
## 15   -0.573211515 2.809099e-02           0.14141414         0        NA
## 16   -0.581756323 2.897411e-02           0.15151515         0        NA
## 17   -0.584304437 2.718049e-02           0.16161616         0        NA
## 18   -0.580213586 2.315554e-02           0.17171717         0        NA
## 19   -0.568934346 1.762201e-02           0.18181818         0        NA
## 20   -0.549987977 1.148311e-02           0.19191919         0        NA
## 21   -0.522910715 5.738708e-03           0.20202020         0        NA
## 22   -0.487092216 1.450842e-03           0.21212121         0        NA
## 23   -0.440789158 1.541379e-04           0.22222222         0        NA
## 24   -0.382751000 2.292146e-03           0.23232323         0        NA
## 25   -0.317738694 4.427089e-03           0.24242424         0        NA
## 26   -0.248647321 5.043925e-03           0.25252525         0        NA
## 27   -0.178015022 3.722501e-03           0.26262626         0        NA
## 28   -0.108519663 1.194379e-03           0.27272727         0        NA
## 29   -0.044006518 2.930805e-04           0.28282828         0        NA
## 30    0.012508779 4.886718e-03           0.29292929         0        NA
## 31    0.062616567 1.518785e-02           0.30303030         0        NA
## 32    0.106184167 3.219325e-02           0.31313131         0        NA
## 33    0.142674750 5.707815e-02           0.32323232         0        NA
## 34    0.171439253 9.102767e-02           0.33333333         0        NA
## 35    0.191809531 1.351565e-01           0.34343434         0        NA
## 36    0.203143616 1.904431e-01           0.35353535         0        NA
## 37    0.204852626 2.576687e-01           0.36363636         0        NA
## 38    0.196418582 3.373622e-01           0.37373737         0        NA
## 39    0.177406799 4.297508e-01           0.38383838         0        NA
## 40    0.147474532 5.347215e-01           0.39393939         0        NA
## 41    0.106376726 6.517921e-01           0.40404040         0        NA
## 42    0.053969401 7.800968e-01           0.41414141         0        NA
## 43   -0.009788991 9.183848e-01           0.42424242         0        NA
## 44   -0.084837977 1.065034e+00           0.43434343         0        NA
## 45   -0.171017008 1.218076e+00           0.44444444         0        NA
## 46   -0.268068741 1.375242e+00           0.45454545         0        NA
## 47   -0.375643449 1.534008e+00           0.46464646         0        NA
## 48   -0.493304372 1.691662e+00           0.47474747         0        NA
## 49   -0.620533828 1.845369e+00           0.48484848         0        NA
## 50   -0.756739899 1.992246e+00           0.49494949         0        NA
## 51   -0.901263537 2.129436e+00           0.50505051         0        NA
## 52   -1.053385923 2.254187e+00           0.51515152         0        NA
## 53   -1.212335967 2.363918e+00           0.52525253         0        NA
## 54   -1.377297797 2.456292e+00           0.53535354         0        NA
## 55   -1.547418171 2.529272e+00           0.54545455         0        NA
## 56   -1.721813707 2.581174e+00           0.55555556         0        NA
## 57   -1.899577876 2.610713e+00           0.56565657         0        NA
## 58   -2.079787726 2.617030e+00           0.57575758         0        NA
## 59   -2.261510290 2.599721e+00           0.58585859         0        NA
## 60   -2.443808690 2.558844e+00           0.59595960         0        NA
## 61   -2.625747923 2.494921e+00           0.60606061         0        NA
## 62   -2.806400343 2.408933e+00           0.61616162         0        NA
## 63   -2.984850873 2.302299e+00           0.62626263         0        NA
## 64   -3.160201948 2.176853e+00           0.63636364         0        NA
## 65   -3.331578234 2.034809e+00           0.64646465         0        NA
## 66   -3.498131135 1.878717e+00           0.65656566         0        NA
## 67   -3.659043103 1.711424e+00           0.66666667         0        NA
## 68   -3.813531778 1.536015e+00           0.67676768         0        NA
## 69   -3.960853925 1.355761e+00           0.68686869         0        NA
## 70   -4.100309181 1.174061e+00           0.69696970         0        NA
## 71   -4.231243556 9.943801e-01           0.70707071         0        NA
## 72   -4.353052630 8.201909e-01           0.71717172         0        NA
## 73   -4.465184336 6.549128e-01           0.72727273         0        NA
## 74   -4.567141163 5.018553e-01           0.73737374         0        NA
## 75   -4.658481493 3.641650e-01           0.74747475         0        NA
## 76   -4.738819531 2.447799e-01           0.75757576         0        NA
## 77   -4.807822759 1.463944e-01           0.76767677         0        NA
## 78   -4.865204224 7.144482e-02           0.77777778         0        NA
## 79   -4.910701243 2.214052e-02           0.78787879         0        NA
## 80   -4.943995174 6.783098e-04           0.79797980         0        NA
## 81   -4.964403447 1.015011e-02           0.80808081         0        NA
## 82   -4.972591614 4.934522e-02           0.81818182         0        NA
## 83   -4.969319582 1.165216e-01           0.82828283         0        NA
## 84   -4.955143056 2.103431e-01           0.83838384         0        NA
## 85   -4.930611929 3.293095e-01           0.84848485         0        NA
## 86   -4.896299930 4.716944e-01           0.85858586         0        NA
## 87   -4.852812003 6.355529e-01           0.86868687         0        NA
## 88   -4.800784425 8.187520e-01           0.87878788         0        NA
## 89   -4.740881754 1.019011e+00           0.88888889         0        NA
## 90   -4.673792158 1.233949e+00           0.89898990         0        NA
## 91   -4.600221865 1.461125e+00           0.90909091         0        NA
## 92   -4.520889139 1.698092e+00           0.91919192         0        NA
## 93   -4.436518036 1.942433e+00           0.92929293         0        NA
## 94   -4.347832147 2.191802e+00           0.93939394         0        NA
## 95   -4.255548442 2.443961e+00           0.94949495         0        NA
## 96   -4.160371353 2.696804e+00           0.95959596         0        NA
## 97   -4.062987188 2.948388e+00           0.96969697         0        NA
## 98   -3.964058945 3.196944e+00           0.97979798         0        NA
## 99   -3.864221595 3.440896e+00           0.98989899         0        NA
## 100  -3.764077898 3.678864e+00           1.00000000         0        NA
## 101  -0.444948730 1.015665e+00           0.00000000         1        NA
## 102  -0.425177012 7.796739e-01           0.01010101         1        NA
## 103  -0.411840762 5.764428e-01           0.02020202         1        NA
## 104  -0.404784852 4.065407e-01           0.03030303         1        NA
## 105  -0.403716377 2.694843e-01           0.04040404         1        NA
## 106  -0.408212466 1.637939e-01           0.05050505         1        NA
## 107  -0.417737294 8.711779e-02           0.06060606         1        NA
## 108  -0.431674611 3.643189e-02           0.07070707         1        NA
## 109  -0.449397907 8.357952e-03           0.08080808         1        NA
## 110  -0.470565425 1.014904e-04           0.09090909         1        NA
## 111  -0.495088350 8.999078e-03           0.10101010         1        NA
## 112  -0.520316620 2.502527e-02           0.11111111         1        NA
## 113  -0.544294888 4.191722e-02           0.12121212         1        NA
## 114  -0.565373425 5.562684e-02           0.13131313         1        NA
## 115  -0.582073817 6.375707e-02           0.14141414         1        NA
## 116  -0.593083388 6.535059e-02           0.15151515         1        NA
## 117  -0.597271731 6.067386e-02           0.16161616         1        NA
## 118  -0.593709700 5.095315e-02           0.17171717         1        NA
## 119  -0.581681949 3.807178e-02           0.18181818         1        NA
## 120  -0.560683981 2.425682e-02           0.19191919         1        NA
## 121  -0.530385217 1.179775e-02           0.20202020         1        NA
## 122  -0.490493423 2.883509e-03           0.21212121         1        NA
## 123  -0.439879355 2.973896e-04           0.22222222         1        NA
## 124  -0.378039557 4.542770e-03           0.23232323         1        NA
## 125  -0.310023742 8.875406e-03           0.24242424         1        NA
## 126  -0.239355545 1.015337e-02           0.25252525         1        NA
## 127  -0.169345824 7.470760e-03           0.26262626         1        NA
## 128  -0.103514030 2.365253e-03           0.27272727         1        NA
## 129  -0.046569024 5.737734e-04           0.28282828         1        NA
## 130  -0.002351282 9.942236e-03           0.29292929         1        NA
## 131   0.029995968 3.173334e-02           0.30303030         1        NA
## 132   0.049678681 6.871041e-02           0.31313131         1        NA
## 133   0.055585508 1.239010e-01           0.32323232         1        NA
## 134   0.046592237 2.001568e-01           0.33333333         1        NA
## 135   0.021664875 2.998768e-01           0.34343434         1        NA
## 136  -0.020087926 4.247699e-01           0.35353535         1        NA
## 137  -0.079383644 5.756534e-01           0.36363636         1        NA
## 138  -0.156746475 7.522970e-01           0.37373737         1        NA
## 139  -0.252495817 9.533220e-01           0.38383838         1        NA
## 140  -0.366740628 1.176163e+00           0.39393939         1        NA
## 141  -0.499378335 1.417096e+00           0.40404040         1        NA
## 142  -0.650097351 1.671329e+00           0.41414141         1        NA
## 143  -0.818382359 1.933157e+00           0.42424242         1        NA
## 144  -1.003521687 2.196158e+00           0.43434343         1        NA
## 145  -1.204616146 2.453437e+00           0.44444444         1        NA
## 146  -1.420588840 2.697890e+00           0.45454545         1        NA
## 147  -1.650195528 2.922483e+00           0.46464646         1        NA
## 148  -1.892035271 3.120532e+00           0.47474747         1        NA
## 149  -2.144561170 3.285966e+00           0.48484848         1        NA
## 150  -2.406091175 3.413577e+00           0.49494949         1        NA
## 151  -2.674819010 3.499222e+00           0.50505051         1        NA
## 152  -2.948825395 3.540001e+00           0.51515152         1        NA
## 153  -3.226089805 3.534381e+00           0.52525253         1        NA
## 154  -3.504503067 3.482267e+00           0.53535354         1        NA
## 155  -3.781881138 3.385026e+00           0.54545455         1        NA
## 156  -4.055980399 3.245446e+00           0.55555556         1        NA
## 157  -4.324514756 3.067651e+00           0.56565657         1        NA
## 158  -4.585174798 2.856949e+00           0.57575758         1        NA
## 159  -4.835649169 2.619634e+00           0.58585859         1        NA
## 160  -5.073648168 2.362747e+00           0.59595960         1        NA
## 161  -5.296929505 2.093787e+00           0.60606061         1        NA
## 162  -5.503325944 1.820413e+00           0.61616162         1        NA
## 163  -5.690774434 1.550119e+00           0.62626263         1        NA
## 164  -5.857346160 1.289915e+00           0.63636364         1        NA
## 165  -6.001276769 1.046037e+00           0.64646465         1        NA
## 166  -6.120995887 8.236791e-01           0.65656566         1        NA
## 167  -6.215154827 6.267893e-01           0.66666667         1        NA
## 168  -6.282651213 4.579336e-01           0.67676768         1        NA
## 169  -6.322648887 3.182359e-01           0.68686869         1        NA
## 170  -6.334590917 2.074088e-01           0.69696970         1        NA
## 171  -6.318202223 1.238719e-01           0.70707071         1        NA
## 172  -6.273475192 6.495760e-02           0.71717172         1        NA
## 173  -6.200621996 2.719951e-02           0.72727273         1        NA
## 174  -6.099938772 6.719906e-03           0.73737374         1        NA
## 175  -5.971199355 1.078762e-14           0.74747475         1        NA
## 176  -5.812933721 4.002599e-03           0.75757576         1        NA
## 177  -5.631362198 8.588462e-03           0.76767677         1        NA
## 178  -5.431430460 9.091875e-03           0.77777778         1        NA
## 179  -5.217539116 5.113544e-03           0.78787879         1        NA
## 180  -4.994355514 2.983417e-04           0.79797980         1        NA
## 181  -4.767926568 4.403168e-03           0.80808081         1        NA
## 182  -4.539828962 2.220251e-02           0.81818182         1        NA
## 183  -4.311409156 5.713974e-02           0.82828283         1        NA
## 184  -4.084768956 1.136539e-01           0.83838384         1        NA
## 185  -3.862048961 1.960721e-01           0.84848485         1        NA
## 186  -3.645284006 3.082074e-01           0.85858586         1        NA
## 187  -3.436335679 4.530611e-01           0.86868687         1        NA
## 188  -3.236852649 6.325994e-01           0.87878788         1        NA
## 189  -3.048246001 8.476075e-01           0.88888889         1        NA
## 190  -2.871674955 1.097624e+00           0.89898990         1        NA
## 191  -2.708040852 1.380953e+00           0.90909091         1        NA
## 192  -2.557988155 1.694751e+00           0.91919192         1        NA
## 193  -2.421911524 2.035175e+00           0.92929293         1        NA
## 194  -2.299968093 2.397574e+00           0.93939394         1        NA
## 195  -2.192094139 2.776721e+00           0.94949495         1        NA
## 196  -2.098025272 3.167050e+00           0.95959596         1        NA
## 197  -2.017319332 3.562901e+00           0.96969697         1        NA
## 198  -1.949381125 3.958738e+00           0.97979798         1        NA
## 199  -1.893488223 4.349353e+00           0.98989899         1        NA
## 200  -1.848817052 4.730028e+00           1.00000000         1        NA
## 201  -1.129314188 3.288345e+00           0.00000000         2        NA
## 202  -1.027461292 2.839801e+00           0.01010101         2        NA
## 203  -0.924943727 2.354229e+00           0.02020202         2        NA
## 204  -0.825081197 1.851290e+00           0.03030303         2        NA
## 205  -0.731436585 1.357009e+00           0.04040404         2        NA
## 206  -0.647583603 9.015996e-01           0.05050505         2        NA
## 207  -0.576842685 5.158754e-01           0.06060606         2        NA
## 208  -0.522015564 2.266928e-01           0.07070707         2        NA
## 209  -0.485174989 5.236793e-02           0.08080808         2        NA
## 210  -0.467789556 5.621313e-04           0.09090909         2        NA
## 211  -0.470433394 6.459775e-02           0.10101010         2        NA
## 212  -0.488826889 2.015562e-01           0.11111111         2        NA
## 213  -0.518437353 3.659769e-01           0.12121212         2        NA
## 214  -0.554212904 5.142403e-01           0.13131313         2        NA
## 215  -0.590720180 6.112333e-01           0.14141414         2        NA
## 216  -0.622512741 6.365401e-01           0.15151515         2        NA
## 217  -0.644557724 5.876896e-01           0.16161616         2        NA
## 218  -0.652660559 4.793831e-01           0.17171717         2        NA
## 219  -0.643842115 3.387298e-01           0.18181818         2        NA
## 220  -0.616628805 1.976362e-01           0.19191919         2        NA
## 221  -0.571221620 8.438309e-02           0.20202020         2        NA
## 222  -0.509510497 1.687328e-02           0.21212121         2        NA
## 223  -0.434793049 1.397016e-03           0.22222222         2        NA
## 224  -0.351862077 2.606923e-02           0.23232323         2        NA
## 225  -0.267682963 5.415410e-02           0.24242424         2        NA
## 226  -0.189397508 6.226882e-02           0.25252525         2        NA
## 227  -0.124076162 4.389530e-02           0.26262626         2        NA
## 228  -0.078359553 1.244745e-02           0.27272727         2        NA
## 229  -0.058812668 2.723043e-03           0.28282828         2        NA
## 230  -0.069599529 5.655520e-02           0.29292929         2        NA
## 231  -0.108806479 1.928222e-01           0.30303030         2        NA
## 232  -0.174389802 4.242541e-01           0.31313131         2        NA
## 233  -0.263246822 7.512533e-01           0.32323232         2        NA
## 234  -0.371284175 1.161259e+00           0.33333333         2        NA
## 235  -0.493719785 1.631420e+00           0.34343434         2        NA
## 236  -0.625445077 2.133115e+00           0.35353535         2        NA
## 237  -0.761383168 2.636948e+00           0.36363636         2        NA
## 238  -0.896802724 3.117013e+00           0.37373737         2        NA
## 239  -1.027561423 3.553709e+00           0.38383838         2        NA
## 240  -1.150265404 3.934874e+00           0.39393939         2        NA
## 241  -1.262342470 4.255439e+00           0.40404040         2        NA
## 242  -1.362036399 4.516096e+00           0.41414141         2        NA
## 243  -1.448336919 4.721488e+00           0.42424242         2        NA
## 244  -1.520864473 4.878403e+00           0.43434343         2        NA
## 245  -1.579730834 4.994244e+00           0.44444444         2        NA
## 246  -1.625396374 5.075925e+00           0.45454545         2        NA
## 247  -1.658542841 5.129181e+00           0.46464646         2        NA
## 248  -1.679977352 5.158202e+00           0.47474747         2        NA
## 249  -1.690579465 5.165492e+00           0.48484848         2        NA
## 250  -1.691298854 5.151837e+00           0.49494949         2        NA
## 251  -1.683206329 5.116320e+00           0.50505051         2        NA
## 252  -1.667595689 5.056352e+00           0.51515152         2        NA
## 253  -1.646128082 4.967755e+00           0.52525253         2        NA
## 254  -1.621004114 4.844953e+00           0.53535354         2        NA
## 255  -1.595142308 4.681379e+00           0.54545455         2        NA
## 256  -1.572336091 4.470207e+00           0.55555556         2        NA
## 257  -1.557356453 4.205462e+00           0.56565657         2        NA
## 258  -1.555964928 3.883504e+00           0.57575758         2        NA
## 259  -1.574803094 3.504724e+00           0.58585859         2        NA
## 260  -1.621131603 3.075130e+00           0.59595960         2        NA
## 261  -1.702404632 2.607362e+00           0.60606061         2        NA
## 262  -1.825684453 2.120666e+00           0.61616162         2        NA
## 263  -1.996924357 1.639434e+00           0.62626263         2        NA
## 264  -2.220174024 1.190279e+00           0.63636364         2        NA
## 265  -2.496786083 7.980029e-01           0.64646465         2        NA
## 266  -2.824722273 4.813207e-01           0.65656566         2        NA
## 267  -3.198069358 2.494435e-01           0.66666667         2        NA
## 268  -3.606880463 1.005985e-01           0.67676768         2        NA
## 269  -4.037482432 2.316028e-02           0.68686869         2        NA
## 270  -4.473757282 0.000000e+00           0.69696970         2        NA
## 271  -4.898867836 1.210997e-02           0.70707071         2        NA
## 272  -5.285129904 2.401856e-02           0.71717172         2        NA
## 273  -5.607138698 2.207144e-02           0.72727273         2        NA
## 274  -5.842529864 9.514019e-03           0.73737374         2        NA
## 275  -5.971199355 0.000000e+00           0.74747475         2        NA
## 276  -5.974606769 9.750020e-03           0.75757576         2        NA
## 277  -5.865435321 2.343632e-02           0.76767677         2        NA
## 278  -5.658282034 2.713664e-02           0.77777778         2        NA
## 279  -5.370632247 1.611479e-02           0.78787879         2        NA
## 280  -5.023536988 9.300061e-04           0.79797980         2        NA
## 281  -4.641575308 1.515596e-02           0.80808081         2        NA
## 282  -4.243407303 8.205219e-02           0.81818182         2        NA
## 283  -3.846041349 2.183977e-01           0.82828283         2        NA
## 284  -3.465150652 4.370326e-01           0.83838384         2        NA
## 285  -3.113406158 7.412981e-01           0.84848485         2        NA
## 286  -2.799919177 1.123494e+00           0.85858586         2        NA
## 287  -2.530141314 1.566032e+00           0.86868687         2        NA
## 288  -2.306107780 2.044662e+00           0.87878788         2        NA
## 289  -2.126929365 2.532717e+00           0.88888889         2        NA
## 290  -1.989436072 3.005252e+00           0.89898990         2        NA
## 291  -1.888879663 3.442180e+00           0.90909091         2        NA
## 292  -1.819615314 3.829949e+00           0.91919192         2        NA
## 293  -1.775702079 4.161770e+00           0.92929293         2        NA
## 294  -1.751384052 4.436734e+00           0.93939394         2        NA
## 295  -1.741435842 4.658296e+00           0.94949495         2        NA
## 296  -1.741374350 4.832604e+00           0.95959596         2        NA
## 297  -1.747552443 4.967007e+00           0.96969697         2        NA
## 298  -1.757158356 5.068942e+00           0.97979798         2        NA
## 299  -1.768147873 5.145219e+00           0.98989899         2        NA
## 300  -1.779135548 5.201671e+00           1.00000000         2        NA
## 301  -1.054939874 2.909700e+00           0.00000000         3        NA
## 302  -0.960604653 2.492670e+00           0.01010101         3        NA
## 303  -0.866852971 2.049227e+00           0.02020202         3        NA
## 304  -0.776656851 1.597758e+00           0.03030303         3        NA
## 305  -0.693125133 1.161249e+00           0.04040404         3        NA
## 306  -0.619302172 7.651756e-01           0.05050505         3        NA
## 307  -0.557946655 4.344083e-01           0.06060606         3        NA
## 308  -0.511316366 1.895585e-01           0.07070707         3        NA
## 309  -0.481007189 4.354977e-02           0.08080808         3        NA
## 310  -0.468095947 4.672306e-04           0.09090909         3        NA
## 311  -0.472908357 5.302219e-02           0.10101010         3        NA
## 312  -0.491465158 1.641874e-01           0.11111111         3        NA
## 313  -0.519629769 2.962432e-01           0.12121212         3        NA
## 314  -0.552854048 4.140223e-01           0.13131313         3        NA
## 315  -0.586284413 4.899120e-01           0.14141414         3        NA
## 316  -0.615072596 5.083913e-01           0.15151515         3        NA
## 317  -0.634737961 4.682013e-01           0.16161616         3        NA
## 318  -0.641529852 3.814187e-01           0.17171717         3        NA
## 319  -0.632751325 2.695512e-01           0.18181818         3        NA
## 320  -0.607010450 1.575947e-01           0.19191919         3        NA
## 321  -0.564368884 6.760708e-02           0.20202020         3        NA
## 322  -0.506353854 1.365066e-02           0.21212121         3        NA
## 323  -0.435662409 1.145293e-03           0.22222222         3        NA
## 324  -0.356370939 2.111159e-02           0.23232323         3        NA
## 325  -0.274917504 4.374326e-02           0.24242424         3        NA
## 326  -0.197832774 5.035164e-02           0.25252525         3        NA
## 327  -0.131627852 3.563818e-02           0.26262626         3        NA
## 328  -0.082515498 1.018803e-02           0.27272727         3        NA
## 329  -0.056785416 2.246965e-03           0.28282828         3        NA
## 330  -0.058619628 4.638347e-02           0.29292929         3        NA
## 331  -0.086592814 1.584171e-01           0.30303030         3        NA
## 332  -0.139276671 3.502668e-01           0.31313131         3        NA
## 333  -0.214299575 6.245223e-01           0.32323232         3        NA
## 334  -0.308351105 9.733467e-01           0.33333333         3        NA
## 335  -0.417411150 1.379996e+00           0.34343434         3        NA
## 336  -0.537045257 1.821964e+00           0.35353535         3        NA
## 337  -0.662710324 2.274790e+00           0.36363636         3        NA
## 338  -0.790035372 2.715598e+00           0.37373737         3        NA
## 339  -0.915053223 3.125736e+00           0.38383838         3        NA
## 340  -1.034368539 3.492189e+00           0.39393939         3        NA
## 341  -1.145256696 3.807806e+00           0.40404040         3        NA
## 342  -1.245696054 4.070605e+00           0.41414141         3        NA
## 343  -1.334342675 4.282518e+00           0.42424242         3        NA
## 344  -1.410461215 4.447958e+00           0.43434343         3        NA
## 345  -1.473828348 4.572472e+00           0.44444444         3        NA
## 346  -1.524625853 4.661651e+00           0.45454545         3        NA
## 347  -1.563339662 4.720346e+00           0.46464646         3        NA
## 348  -1.590679078 4.752173e+00           0.47474747         3        NA
## 349  -1.607527306 4.759237e+00           0.48484848         3        NA
## 350  -1.614930712 4.742017e+00           0.49494949         3        NA
## 351  -1.614129939 4.699358e+00           0.50505051         3        NA
## 352  -1.606631214 4.628573e+00           0.51515152         3        NA
## 353  -1.594311056 4.525658e+00           0.52525253         3        NA
## 354  -1.579542127 4.385705e+00           0.53535354         3        NA
## 355  -1.565322533 4.203532e+00           0.54545455         3        NA
## 356  -1.555385947 3.974590e+00           0.55555556         3        NA
## 357  -1.554266151 3.696104e+00           0.56565657         3        NA
## 358  -1.567287924 3.368337e+00           0.57575758         3        NA
## 359  -1.600457665 2.995743e+00           0.58585859         3        NA
## 360  -1.660232444 2.587742e+00           0.59595960         3        NA
## 361  -1.753156072 2.158779e+00           0.60606061         3        NA
## 362  -1.885365091 1.727442e+00           0.61616162         3        NA
## 363  -2.061985570 1.314591e+00           0.62626263         3        NA
## 364  -2.286461632 9.406953e-01           0.63636364         3        NA
## 365  -2.559876328 6.228364e-01           0.64646465         3        NA
## 366  -2.880341925 3.720552e-01           0.65656566         3        NA
## 367  -3.242547614 1.916972e-01           0.66666667         3        NA
## 368  -3.637558220 7.726054e-02           0.67676768         3        NA
## 369  -4.052971968 1.791738e-02           0.68686869         3        NA
## 370  -4.473757282 0.000000e+00           0.69696970         3        NA
## 371  -4.883481498 8.987342e-03           0.70707071         3        NA
## 372  -5.258768461 1.693973e-02           0.71717172         3        NA
## 373  -5.577459979 1.475321e-02           0.72727273         3        NA
## 374  -5.819889648 5.946014e-03           0.73737374         3        NA
## 375  -5.971199355 0.000000e+00           0.74747475         3        NA
## 376  -6.020682902 0.000000e+00           0.75757576         3        NA
## 377  -5.926156928 5.291856e-03           0.76767677         3        NA
## 378  -5.713003453 1.104266e-02           0.77777778         3        NA
## 379  -5.405734075 8.722501e-03           0.78787879         3        NA
## 380  -5.029820736 6.322887e-04           0.79797980         3        NA
## 381  -4.616638122 1.029933e-02           0.80808081         3        NA
## 382  -4.184327744 5.550433e-02           0.81818182         3        NA
## 383  -3.749483909 1.493122e-01           0.82828283         3        NA
## 384  -3.329247552 3.040200e-01           0.83838384         3        NA
## 385  -2.938325464 5.268104e-01           0.84848485         3        NA
## 386  -2.588022893 8.176710e-01           0.85858586         3        NA
## 387  -2.285848006 1.168855e+00           0.86868687         3        NA
## 388  -2.035527995 1.565890e+00           0.87878788         3        NA
## 389  -1.837345907 1.989811e+00           0.88888889         3        NA
## 390  -1.688706845 2.420004e+00           0.89898990         3        NA
## 391  -1.584838778 2.837027e+00           0.90909091         3        NA
## 392  -1.519537650 3.224832e+00           0.91919192         3        NA
## 393  -1.485879309 3.572086e+00           0.92929293         3        NA
## 394  -1.476839679 3.872524e+00           0.93939394         3        NA
## 395  -1.485786210 4.124506e+00           0.94949495         3        NA
## 396  -1.506824869 4.330046e+00           0.95959596         3        NA
## 397  -1.535005222 4.493626e+00           0.96969697         3        NA
## 398  -1.566399935 4.621051e+00           0.97979798         3        NA
## 399  -1.598083593 4.718509e+00           0.98989899         3        NA
## 400  -1.628039454 4.791908e+00           1.00000000         3        NA
## 401  -1.021303887 2.629460e+00           0.00000000         4        NA
## 402  -0.930789202 2.246495e+00           0.01010101         4        NA
## 403  -0.841252174 1.841666e+00           0.02020202         4        NA
## 404  -0.755517792 1.431838e+00           0.03030303         4        NA
## 405  -0.676514945 1.037716e+00           0.04040404         4        NA
## 406  -0.607086924 6.819022e-01           0.05050505         4        NA
## 407  -0.549785506 3.861349e-01           0.06060606         4        NA
## 408  -0.506672822 1.681075e-01           0.07070707         4        NA
## 409  -0.479176295 3.855390e-02           0.08080808         4        NA
## 410  -0.468234020 4.136418e-04           0.09090909         4        NA
## 411  -0.474082465 4.673659e-02           0.10101010         4        NA
## 412  -0.492871170 1.443522e-01           0.11111111         4        NA
## 413  -0.520619961 2.599035e-01           0.12121212         4        NA
## 414  -0.552972732 3.625883e-01           0.13131313         4        NA
## 415  -0.585294033 4.284233e-01           0.14141414         4        NA
## 416  -0.612959917 4.440813e-01           0.15151515         4        NA
## 417  -0.631697669 4.086633e-01           0.16161616         4        NA
## 418  -0.637926048 3.328027e-01           0.17171717         4        NA
## 419  -0.629059847 2.352337e-01           0.18181818         4        NA
## 420  -0.603747017 1.376440e-01           0.19191919         4        NA
## 421  -0.562009415 5.915205e-02           0.20202020         4        NA
## 422  -0.505253383 1.198482e-02           0.21212121         4        NA
## 423  -0.435969497 1.010173e-03           0.22222222         4        NA
## 424  -0.357981385 1.854243e-02           0.23232323         4        NA
## 425  -0.277526130 3.838473e-02           0.24242424         4        NA
## 426  -0.200901529 4.419769e-02           0.25252525         4        NA
## 427  -0.134399610 3.132413e-02           0.26262626         4        NA
## 428  -0.084055300 8.979068e-03           0.27272727         4        NA
## 429  -0.056025517 1.985682e-03           0.28282828         4        NA
## 430  -0.054465421 4.089602e-02           0.29292929         4        NA
## 431  -0.078111583 1.397295e-01           0.30303030         4        NA
## 432  -0.125737461 3.094027e-01           0.31313131         4        NA
## 433  -0.195218066 5.528508e-01           0.32323232         4        NA
## 434  -0.283515647 8.639051e-01           0.33333333         4        NA
## 435  -0.386884979 1.228442e+00           0.34343434         4        NA
## 436  -0.501145682 1.626985e+00           0.35353535         4        NA
## 437  -0.621968697 2.037956e+00           0.36363636         4        NA
## 438  -0.745143533 2.440792e+00           0.37373737         4        NA
## 439  -0.866803066 2.818336e+00           0.38383838         4        NA
## 440  -0.983591368 3.158225e+00           0.39393939         4        NA
## 441  -1.092768333 3.453230e+00           0.40404040         4        NA
## 442  -1.192252333 3.700769e+00           0.41414141         4        NA
## 443  -1.280608356 3.901891e+00           0.42424242         4        NA
## 444  -1.356993658 4.060040e+00           0.43434343         4        NA
## 445  -1.421075667 4.179857e+00           0.44444444         4        NA
## 446  -1.472937963 4.266176e+00           0.45454545         4        NA
## 447  -1.512989572 4.323273e+00           0.46464646         4        NA
## 448  -1.541891078 4.354374e+00           0.47474747         4        NA
## 449  -1.560508270 4.361362e+00           0.48484848         4        NA
## 450  -1.569900555 4.344654e+00           0.49494949         4        NA
## 451  -1.571347287 4.303184e+00           0.50505051         4        NA
## 452  -1.566410607 4.234513e+00           0.51515152         4        NA
## 453  -1.557028507 4.135065e+00           0.52525253         4        NA
## 454  -1.545626667 4.000537e+00           0.53535354         4        NA
## 455  -1.535232630 3.826531e+00           0.54545455         4        NA
## 456  -1.529571277 3.609401e+00           0.55555556         4        NA
## 457  -1.533117215 3.347290e+00           0.56565657         4        NA
## 458  -1.551078256 3.041229e+00           0.57575758         4        NA
## 459  -1.589285490 2.696084e+00           0.58585859         4        NA
## 460  -1.653970399 2.321117e+00           0.59595960         4        NA
## 461  -1.751418393 1.929884e+00           0.60606061         4        NA
## 462  -1.887501158 1.539316e+00           0.61616162         4        NA
## 463  -2.067106527 1.167966e+00           0.62626263         4        NA
## 464  -2.293502904 8.336439e-01           0.63636364         4        NA
## 465  -2.567693448 5.508628e-01           0.64646465         4        NA
## 466  -2.887830771 3.286496e-01           0.65656566         4        NA
## 467  -3.248773704 1.692760e-01           0.66666667         4        NA
## 468  -3.641873917 6.827809e-02           0.67676768         4        NA
## 469  -4.055094609 1.587027e-02           0.68686869         4        NA
## 470  -4.473757282 4.817201e-13           0.69696970         4        NA
## 471  -4.881661525 7.971981e-03           0.70707071         4        NA
## 472  -5.256063643 1.502522e-02           0.71717172         4        NA
## 473  -5.575121524 1.311336e-02           0.72727273         4        NA
## 474  -5.819023306 5.307658e-03           0.73737374         4        NA
## 475  -5.971199355 8.530624e-13           0.74747475         4        NA
## 476  -6.020682902 2.546573e-13           0.75757576         4        NA
## 477  -5.961019152 1.184196e-12           0.76767677         4        NA
## 478  -5.752448372 3.782343e-03           0.77777778         4        NA
## 479  -5.431833266 5.118904e-03           0.78787879         4        NA
## 480  -5.034385156 4.831173e-04           0.79797980         4        NA
## 481  -4.599635420 8.036805e-03           0.80808081         4        NA
## 482  -4.145288733 4.346123e-02           0.81818182         4        NA
## 483  -3.687032974 1.180135e-01           0.82828283         4        NA
## 484  -3.242673383 2.429563e-01           0.83838384         4        NA
## 485  -2.827957906 4.259718e-01           0.84848485         4        NA
## 486  -2.455343469 6.691767e-01           0.85858586         4        NA
## 487  -2.133437446 9.682381e-01           0.86868687         4        NA
## 488  -1.866893061 1.312735e+00           0.87878788         4        NA
## 489  -1.656661423 1.687625e+00           0.88888889         4        NA
## 490  -1.500510116 2.075442e+00           0.89898990         4        NA
## 491  -1.393713457 2.458687e+00           0.90909091         4        NA
## 492  -1.329820441 2.821951e+00           0.91919192         4        NA
## 493  -1.301416015 3.153386e+00           0.92929293         4        NA
## 494  -1.300807990 3.445403e+00           0.93939394         4        NA
## 495  -1.320592852 3.694622e+00           0.94949495         4        NA
## 496  -1.354075567 3.901262e+00           0.95959596         4        NA
## 497  -1.395538509 4.068217e+00           0.96969697         4        NA
## 498  -1.440370802 4.200054e+00           0.97979798         4        NA
## 499  -1.485080519 4.302103e+00           0.98989899         4        NA
## 500  -1.527218247 4.379758e+00           1.00000000         4        NA
## 501  -1.091373742 2.338732e+00           0.00000000         5        NA
## 502  -0.993821913 2.014540e+00           0.01010101         5        NA
## 503  -0.896062625 1.665607e+00           0.02020202         5        NA
## 504  -0.801249449 1.306195e+00           0.03030303         5        NA
## 505  -0.712738243 9.548315e-01           0.04040404         5        NA
## 506  -0.633868635 6.326986e-01           0.05050505         5        NA
## 507  -0.567718313 3.611012e-01           0.06060606         5        NA
## 508  -0.516858504 1.583182e-01           0.07070707         5        NA
## 509  -0.483163367 3.650692e-02           0.08080808         5        NA
## 510  -0.467938203 3.917789e-04           0.09090909         5        NA
## 511  -0.471653403 4.484213e-02           0.10101010         5        NA
## 512  -0.490190155 1.395738e-01           0.11111111         5        NA
## 513  -0.519208348 2.529131e-01           0.12121212         5        NA
## 514  -0.553889160 3.547500e-01           0.13131313         5        NA
## 515  -0.589060402 4.210407e-01           0.14141414         5        NA
## 516  -0.619540411 4.379568e-01           0.15151515         5        NA
## 517  -0.640537575 4.040023e-01           0.16161616         5        NA
## 518  -0.648049155 3.293925e-01           0.17171717         5        NA
## 519  -0.639216711 2.327452e-01           0.18181818         5        NA
## 520  -0.612600937 1.358785e-01           0.19191919         5        NA
## 521  -0.568343298 5.809947e-02           0.20202020         5        NA
## 522  -0.508180739 1.165309e-02           0.21212121         5        NA
## 523  -0.435161313 9.688958e-04           0.22222222         5        NA
## 524  -0.353779314 1.800929e-02           0.23232323         5        NA
## 525  -0.270767344 3.738065e-02           0.24242424         5        NA
## 526  -0.193003166 4.299724e-02           0.25252525         5        NA
## 527  -0.127313989 3.034999e-02           0.26262626         5        NA
## 528  -0.080148630 8.628944e-03           0.27272727         5        NA
## 529  -0.057933945 1.892689e-03           0.28282828         5        NA
## 530  -0.064820179 3.923505e-02           0.29292929         5        NA
## 531  -0.099101442 1.338545e-01           0.30303030         5        NA
## 532  -0.158983120 2.949969e-01           0.31313131         5        NA
## 533  -0.241657994 5.235634e-01           0.32323232         5        NA
## 534  -0.343351944 8.115047e-01           0.33333333         5        NA
## 535  -0.459598583 1.143488e+00           0.34343434         5        NA
## 536  -0.585576365 1.499886e+00           0.35353535         5        NA
## 537  -0.716447414 1.860165e+00           0.36363636         5        NA
## 538  -0.847659840 2.205872e+00           0.37373737         5        NA
## 539  -0.975188247 2.522689e+00           0.38383838         5        NA
## 540  -1.095698469 2.801352e+00           0.39393939         5        NA
## 541  -1.206632966 3.037556e+00           0.40404040         5        NA
## 542  -1.306222210 3.231131e+00           0.41414141         5        NA
## 543  -1.393434204 3.384842e+00           0.42424242         5        NA
## 544  -1.467878709 3.503137e+00           0.43434343         5        NA
## 545  -1.529684740 3.591052e+00           0.44444444         5        NA
## 546  -1.579369894 3.653387e+00           0.45454545         5        NA
## 547  -1.617718332 3.694154e+00           0.46464646         5        NA
## 548  -1.645681565 3.716268e+00           0.47474747         5        NA
## 549  -1.664312729 3.721393e+00           0.48484848         5        NA
## 550  -1.674741374 3.709887e+00           0.49494949         5        NA
## 551  -1.678191892 3.680804e+00           0.50505051         5        NA
## 552  -1.676044630 3.631943e+00           0.51515152         5        NA
## 553  -1.669934426 3.559973e+00           0.52525253         5        NA
## 554  -1.661876566 3.460680e+00           0.53535354         5        NA
## 555  -1.654405169 3.329413e+00           0.54545455         5        NA
## 556  -1.650703945 3.161757e+00           0.55555556         5        NA
## 557  -1.654704664 2.954461e+00           0.56565657         5        NA
## 558  -1.671125482 2.706525e+00           0.57575758         5        NA
## 559  -1.705420420 2.420282e+00           0.58585859         5        NA
## 560  -1.763614038 2.102221e+00           0.59595960         5        NA
## 561  -1.852002584 1.763260e+00           0.60606061         5        NA
## 562  -1.976715167 1.418207e+00           0.61616162         5        NA
## 563  -2.143145587 1.084312e+00           0.62626263         5        NA
## 564  -2.355286171 7.790463e-01           0.63636364         5        NA
## 565  -2.615017219 5.174701e-01           0.64646465         5        NA
## 566  -2.921426375 3.097990e-01           0.65656566         5        NA
## 567  -3.270248536 1.597781e-01           0.66666667         5        NA
## 568  -3.653527117 6.436076e-02           0.67676768         5        NA
## 569  -4.059610925 1.488602e-02           0.68686869         5        NA
## 570  -4.473757300 1.270848e-10           0.69696970         5        NA
## 571  -4.879146549 7.440768e-03           0.70707071         5        NA
## 572  -5.252940755 1.401848e-02           0.71717172         5        NA
## 573  -5.572755459 1.218297e-02           0.72727273         5        NA
## 574  -5.817987469 4.894001e-03           0.73737374         5        NA
## 575  -5.971199349 0.000000e+00           0.74747475         5        NA
## 576  -6.020682963 0.000000e+00           0.75757576         5        NA
## 577  -5.961018893 0.000000e+00           0.76767677         5        NA
## 578  -5.752802363 3.472846e-03           0.77777778         5        NA
## 579  -5.432206366 4.717539e-03           0.78787879         5        NA
## 580  -5.034481054 4.444604e-04           0.79797980         5        NA
## 581  -4.599100692 7.457486e-03           0.80808081         5        NA
## 582  -4.145234255 4.061175e-02           0.81818182         5        NA
## 583  -3.689881349 1.106667e-01           0.82828283         5        NA
## 584  -3.251519158 2.280089e-01           0.83838384         5        NA
## 585  -2.846082666 3.991528e-01           0.84848485         5        NA
## 586  -2.485758970 6.248731e-01           0.85858586         5        NA
## 587  -2.178495077 8.995600e-01           0.86868687         5        NA
## 588  -1.928001168 1.211892e+00           0.87878788         5        NA
## 589  -1.734136705 1.546595e+00           0.88888889         5        NA
## 590  -1.593568946 1.886820e+00           0.89898990         5        NA
## 591  -1.500589330 2.216540e+00           0.90909091         5        NA
## 592  -1.447978456 2.522492e+00           0.91919192         5        NA
## 593  -1.427826266 2.795350e+00           0.92929293         5        NA
## 594  -1.432237563 3.030047e+00           0.93939394         5        NA
## 595  -1.453880009 3.225409e+00           0.94949495         5        NA
## 596  -1.486358015 3.383306e+00           0.95959596         5        NA
## 597  -1.524418357 3.507635e+00           0.96969697         5        NA
## 598  -1.564009782 3.603329e+00           0.97979798         5        NA
## 599  -1.602228741 3.675564e+00           0.98989899         5        NA
## 600  -1.637186900 3.729208e+00           1.00000000         5        NA
## 601  -1.048034308 2.103775e+00           0.00000000         6        NA
## 602  -0.954830215 1.803264e+00           0.01010101         6        NA
## 603  -0.862155342 1.483322e+00           0.02020202         6        NA
## 604  -0.772960394 1.157205e+00           0.03030303         6        NA
## 605  -0.690336324 8.415417e-01           0.04040404         6        NA
## 606  -0.617313877 5.548224e-01           0.05050505         6        NA
## 607  -0.556642805 3.151510e-01           0.06060606         6        NA
## 608  -0.510576508 1.375837e-01           0.07070707         6        NA
## 609  -0.480710095 3.162039e-02           0.08080808         6        NA
## 610  -0.468119395 3.392546e-04           0.09090909         6        NA
## 611  -0.473128779 3.853126e-02           0.10101010         6        NA
## 612  -0.491789578 1.193744e-01           0.11111111         6        NA
## 613  -0.519988685 2.154756e-01           0.12121212         6        NA
## 614  -0.553198037 3.012474e-01           0.13131313         6        NA
## 615  -0.586581742 3.565664e-01           0.14141414         6        NA
## 616  -0.615306588 3.700968e-01           0.15151515         6        NA
## 617  -0.634904097 3.408900e-01           0.16161616         6        NA
## 618  -0.641632795 2.777236e-01           0.17171717         6        NA
## 619  -0.632802099 1.962634e-01           0.18181818         6        NA
## 620  -0.607023957 1.147290e-01           0.19191919         6        NA
## 621  -0.564361907 4.920175e-02           0.20202020         6        NA
## 622  -0.506343731 9.927954e-03           0.21212121         6        NA
## 623  -0.435667835 8.322331e-04           0.22222222         6        NA
## 624  -0.356409581 1.535292e-02           0.23232323         6        NA
## 625  -0.274992660 3.181641e-02           0.24242424         6        NA
## 626  -0.197935064 3.662029e-02           0.25252525         6        NA
## 627  -0.131733492 2.591244e-02           0.26262626         6        NA
## 628  -0.082582811 7.403803e-03           0.27272727         6        NA
## 629  -0.056745927 1.632041e-03           0.28282828         6        NA
## 630  -0.058380862 3.370259e-02           0.29292929         6        NA
## 631  -0.086063186 1.150934e-01           0.30303030         6        NA
## 632  -0.138356263 2.543940e-01           0.31313131         6        NA
## 633  -0.212879949 4.533768e-01           0.32323232         6        NA
## 634  -0.306318477 7.062256e-01           0.33333333         6        NA
## 635  -0.414651679 1.000671e+00           0.34343434         6        NA
## 636  -0.533451799 1.320303e+00           0.35353535         6        NA
## 637  -0.658190064 1.647352e+00           0.36363636         6        NA
## 638  -0.784517725 1.965265e+00           0.37373737         6        NA
## 639  -0.908497451 2.260603e+00           0.38383838         6        NA
## 640  -1.026770656 2.524054e+00           0.39393939         6        NA
## 641  -1.136655381 2.750562e+00           0.40404040         6        NA
## 642  -1.236177425 2.938811e+00           0.41414141         6        NA
## 643  -1.324043926 3.090289e+00           0.42424242         6        NA
## 644  -1.399573105 3.208246e+00           0.43434343         6        NA
## 645  -1.462596517 3.296708e+00           0.44444444         6        NA
## 646  -1.513350745 3.359689e+00           0.45454545         6        NA
## 647  -1.552374625 3.400624e+00           0.46464646         6        NA
## 648  -1.580425908 3.422009e+00           0.47474747         6        NA
## 649  -1.598428277 3.425206e+00           0.48484848         6        NA
## 650  -1.607455989 3.410358e+00           0.49494949         6        NA
## 651  -1.608759311 3.376402e+00           0.50505051         6        NA
## 652  -1.603829436 3.321163e+00           0.51515152         6        NA
## 653  -1.594496711 3.241555e+00           0.52525253         6        NA
## 654  -1.583050928 3.133933e+00           0.53535354         6        NA
## 655  -1.572367241 2.994628e+00           0.54545455         6        NA
## 656  -1.566016472 2.820672e+00           0.55555556         6        NA
## 657  -1.568334759 2.610664e+00           0.56565657         6        NA
## 658  -1.584425448 2.365678e+00           0.57575758         6        NA
## 659  -1.620066945 2.090019e+00           0.58585859         6        NA
## 660  -1.681504580 1.791609e+00           0.59595960         6        NA
## 661  -1.775113305 1.481806e+00           0.60606061         6        NA
## 662  -1.906931117 1.174517e+00           0.61616162         6        NA
## 663  -2.082080078 8.846528e-01           0.62626263         6        NA
## 664  -2.304111259 6.261345e-01           0.63636364         6        NA
## 665  -2.574329645 4.098434e-01           0.64646465         6        NA
## 666  -2.891172420 2.419797e-01           0.65656566         6        NA
## 667  -3.249726608 1.232564e-01           0.66666667         6        NA
## 668  -3.641479821 4.916944e-02           0.67676768         6        NA
## 669  -4.054414018 1.133810e-02           0.68686869         6        NA
## 670  -4.473757310 0.000000e+00           0.69696970         6        NA
## 671  -4.883123558 4.882873e-03           0.70707071         6        NA
## 672  -5.259147322 7.242792e-03           0.71717172         6        NA
## 673  -5.579280386 3.688525e-03           0.72727273         6        NA
## 674  -5.822608557 0.000000e+00           0.73737374         6        NA
## 675  -5.971199398 4.064317e-08           0.74747475         6        NA
## 676  -6.020682968 0.000000e+00           0.75757576         6        NA
## 677  -5.961018824 3.542978e-08           0.76767677         6        NA
## 678  -5.752443079 3.057265e-03           0.77777778         6        NA
## 679  -5.431803563 4.142689e-03           0.78787879         6        NA
## 680  -5.034379160 3.910340e-04           0.79797980         6        NA
## 681  -4.599641340 6.521739e-03           0.80808081         6        NA
## 682  -4.145872139 3.533323e-02           0.81818182         6        NA
## 683  -3.689258180 9.601744e-02           0.82828283         6        NA
## 684  -3.247825799 1.976738e-01           0.83838384         6        NA
## 685  -2.837347092 3.463680e-01           0.84848485         6        NA
## 686  -2.470126081 5.435106e-01           0.85858586         6        NA
## 687  -2.154471744 7.851704e-01           0.86868687         6        NA
## 688  -1.894638609 1.062435e+00           0.87878788         6        NA
## 689  -1.691132044 1.362700e+00           0.88888889         6        NA
## 690  -1.541280803 1.671542e+00           0.89898990         6        NA
## 691  -1.439974851 1.974733e+00           0.90909091         6        NA
## 692  -1.380469142 2.259979e+00           0.91919192         6        NA
## 693  -1.355165860 2.518085e+00           0.92929293         6        NA
## 694  -1.356306902 2.743448e+00           0.93939394         6        NA
## 695  -1.376531442 2.933924e+00           0.94949495         6        NA
## 696  -1.409276913 3.090244e+00           0.95959596         6        NA
## 697  -1.449022616 3.215199e+00           0.96969697         6        NA
## 698  -1.491391381 3.312794e+00           0.97979798         6        NA
## 699  -1.533135447 3.387503e+00           0.98989899         6        NA
## 700  -1.572037781 3.443727e+00           1.00000000         6        NA
## 701  -1.099019908 1.914772e+00           0.00000000         7        NA
## 702  -1.000775631 1.651103e+00           0.01010101         7        NA
## 703  -0.902164107 1.366630e+00           0.02020202         7        NA
## 704  -0.806376256 1.072942e+00           0.03030303         7        NA
## 705  -0.716818961 7.852048e-01           0.04040404         7        NA
## 706  -0.636892994 5.208688e-01           0.05050505         7        NA
## 707  -0.569742178 2.975839e-01           0.06060606         7        NA
## 708  -0.518002734 1.305917e-01           0.07070707         7        NA
## 709  -0.483606472 3.013556e-02           0.08080808         7        NA
## 710  -0.467906080 3.234376e-04           0.09090909         7        NA
## 711  -0.471401897 3.707975e-02           0.10101010         7        NA
## 712  -0.489943969 1.155268e-01           0.11111111         7        NA
## 713  -0.519148198 2.095118e-01           0.12121212         7        NA
## 714  -0.554138188 2.940797e-01           0.13131313         7        NA
## 715  -0.589674648 3.492385e-01           0.14141414         7        NA
## 716  -0.620506259 3.634398e-01           0.15151515         7        NA
## 717  -0.641776970 3.353749e-01           0.16161616         7        NA
## 718  -0.649432340 2.734884e-01           0.17171717         7        NA
## 719  -0.640581432 1.932427e-01           0.18181818         7        NA
## 720  -0.613776330 1.127886e-01           0.19191919         7        NA
## 721  -0.569176268 4.819788e-02           0.20202020         7        NA
## 722  -0.508562658 9.655184e-03           0.21212121         7        NA
## 723  -0.435056696 8.014157e-04           0.22222222         7        NA
## 724  -0.353239176 1.491969e-02           0.23232323         7        NA
## 725  -0.269904019 3.097772e-02           0.24242424         7        NA
## 726  -0.192000312 3.562696e-02           0.25252525         7        NA
## 727  -0.126419700 2.513421e-02           0.26262626         7        NA
## 728  -0.079658660 7.138492e-03           0.27272727         7        NA
## 729  -0.058171527 1.564102e-03           0.28282828         7        NA
## 730  -0.066100843 3.244753e-02           0.29292929         7        NA
## 731  -0.101680181 1.106686e-01           0.30303030         7        NA
## 732  -0.163037687 2.437338e-01           0.31313131         7        NA
## 733  -0.247275264 4.321790e-01           0.32323232         7        NA
## 734  -0.350522691 6.691224e-01           0.33333333         7        NA
## 735  -0.468221457 9.417061e-01           0.34343434         7        NA
## 736  -0.595470138 1.233617e+00           0.35353535         7        NA
## 737  -0.727369531 1.527916e+00           0.36363636         7        NA
## 738  -0.859328728 1.809499e+00           0.37373737         7        NA
## 739  -0.987306641 2.066764e+00           0.38383838         7        NA
## 740  -1.107975212 2.292322e+00           0.39393939         7        NA
## 741  -1.218801200 2.482876e+00           0.40404040         7        NA
## 742  -1.318052558 2.638500e+00           0.41414141         7        NA
## 743  -1.404742269 2.761626e+00           0.42424242         7        NA
## 744  -1.478526905 2.856003e+00           0.43434343         7        NA
## 745  -1.539579086 2.925799e+00           0.44444444         7        NA
## 746  -1.588452765 2.974931e+00           0.45454545         7        NA
## 747  -1.625958480 3.006626e+00           0.46464646         7        NA
## 748  -1.653062750 3.023171e+00           0.47474747         7        NA
## 749  -1.670822317 3.025795e+00           0.48484848         7        NA
## 750  -1.680360136 3.014628e+00           0.49494949         7        NA
## 751  -1.682886083 2.988707e+00           0.50505051         7        NA
## 752  -1.679761255 2.946034e+00           0.51515152         7        NA
## 753  -1.672600390 2.883695e+00           0.52525253         7        NA
## 754  -1.663402172 2.798112e+00           0.53535354         7        NA
## 755  -1.654692138 2.685457e+00           0.54545455         7        NA
## 756  -1.649657734 2.542271e+00           0.55555556         7        NA
## 757  -1.652250423 2.366270e+00           0.56565657         7        NA
## 758  -1.667226436 2.157259e+00           0.57575758         7        NA
## 759  -1.700096921 1.917972e+00           0.58585859         7        NA
## 760  -1.756961006 1.654648e+00           0.59595960         7        NA
## 761  -1.844202714 1.377058e+00           0.60606061         7        NA
## 762  -1.968045205 1.097862e+00           0.61616162         7        NA
## 763  -2.133973261 8.312200e-01           0.62626263         7        NA
## 764  -2.346056128 5.908691e-01           0.63636364         7        NA
## 765  -2.606225540 3.880118e-01           0.64646465         7        NA
## 766  -2.913584925 2.295227e-01           0.65656566         7        NA
## 767  -3.263842451 1.169435e-01           0.66666667         7        NA
## 768  -3.648971473 4.657361e-02           0.67676768         7        NA
## 769  -4.057217946 1.069520e-02           0.68686869         7        NA
## 770  -4.473757310 2.704558e-08           0.69696970         7        NA
## 771  -4.881767371 4.583244e-03           0.70707071         7        NA
## 772  -5.257703741 6.789288e-03           0.71717172         7        NA
## 773  -5.578529473 3.443626e-03           0.72727273         7        NA
## 774  -5.822608548 2.660690e-08           0.73737374         7        NA
## 775  -5.971199381 6.609013e-08           0.74747475         7        NA
## 776  -6.020682974 0.000000e+00           0.75757576         7        NA
## 777  -5.961018801 5.666348e-08           0.76767677         7        NA
## 778  -5.752659736 2.850222e-03           0.77777778         7        NA
## 779  -5.432020755 3.872588e-03           0.78787879         7        NA
## 780  -5.034434002 3.650896e-04           0.79797980         7        NA
## 781  -4.599334360 6.126426e-03           0.80808081         7        NA
## 782  -4.146064689 3.335830e-02           0.81818182         7        NA
## 783  -3.691750015 9.088083e-02           0.82828283         7        NA
## 784  -3.254879170 1.872040e-01           0.83838384         7        NA
## 785  -2.851327596 3.276584e-01           0.84848485         7        NA
## 786  -2.493168842 5.128535e-01           0.85858586         7        NA
## 787  -2.188201136 7.381360e-01           0.86868687         7        NA
## 788  -1.939971221 9.941351e-01           0.87878788         7        NA
## 789  -1.748180615 1.268219e+00           0.88888889         7        NA
## 790  -1.609361492 1.546464e+00           0.89898990         7        NA
## 791  -1.517705758 1.815656e+00           0.90909091         7        NA
## 792  -1.465937001 2.064905e+00           0.91919192         7        NA
## 793  -1.446131812 2.286611e+00           0.92929293         7        NA
## 794  -1.450421319 2.476726e+00           0.93939394         7        NA
## 795  -1.471531367 2.634422e+00           0.94949495         7        NA
## 796  -1.503146359 2.761382e+00           0.95959596         7        NA
## 797  -1.540104213 2.860927e+00           0.96969697         7        NA
## 798  -1.578446077 2.937201e+00           0.97979798         7        NA
## 799  -1.615353947 2.994507e+00           0.98989899         7        NA
## 800  -1.649012425 3.036860e+00           1.00000000         7        NA
## 801  -1.140595505 1.759148e+00           0.00000000         8        NA
## 802  -1.038395961 1.524306e+00           0.01010101         8        NA
## 803  -0.935029775 1.268112e+00           0.02020202         8        NA
## 804  -0.833885834 1.000788e+00           0.03030303         8        NA
## 805  -0.738639743 7.362233e-01           0.04040404         8        NA
## 806  -0.653013042 4.908656e-01           0.05050505         8        NA
## 807  -0.580495413 2.817968e-01           0.06060606         8        NA
## 808  -0.524061774 1.242035e-01           0.07070707         8        NA
## 809  -0.485943120 2.876112e-02           0.08080808         8        NA
## 810  -0.467737905 3.088761e-04           0.09090909         8        NA
## 811  -0.470101076 3.567392e-02           0.10101010         8        NA
## 812  -0.488703112 1.116575e-01           0.11111111         8        NA
## 813  -0.518917001 2.032778e-01           0.12121212         8        NA
## 814  -0.555555301 2.862766e-01           0.13131313         8        NA
## 815  -0.593020917 3.409241e-01           0.14141414         8        NA
## 816  -0.625694931 3.555886e-01           0.15151515         8        NA
## 817  -0.648384562 3.286733e-01           0.16161616         8        NA
## 818  -0.656768401 2.682795e-01           0.17171717         8        NA
## 819  -0.647791090 1.895808e-01           0.18181818         8        NA
## 820  -0.619966104 1.105390e-01           0.19191919         8        NA
## 821  -0.573551406 4.711285e-02           0.20202020         8        NA
## 822  -0.510564621 9.385180e-03           0.21212121         8        NA
## 823  -0.434508707 7.729352e-04           0.22222222         8        NA
## 824  -0.350413134 1.449501e-02           0.23232323         8        NA
## 825  -0.265393231 3.014058e-02           0.24242424         8        NA
## 826  -0.186766959 3.463981e-02           0.25252525         8        NA
## 827  -0.121757420 2.437682e-02           0.26262626         8        NA
## 828  -0.077105691 6.889301e-03           0.27272727         8        NA
## 829  -0.059409934 1.501964e-03           0.28282828         8        NA
## 830  -0.072773606 3.126299e-02           0.29292929         8        NA
## 831  -0.115107831 1.064836e-01           0.30303030         8        NA
## 832  -0.184139861 2.337545e-01           0.31313131         8        NA
## 833  -0.276505296 4.126476e-01           0.32323232         8        NA
## 834  -0.387845184 6.355418e-01           0.33333333         8        NA
## 835  -0.513137119 8.892956e-01           0.34343434         8        NA
## 836  -0.647082786 1.157899e+00           0.35353535         8        NA
## 837  -0.784485029 1.425295e+00           0.36363636         8        NA
## 838  -0.920572752 1.677700e+00           0.37373737         8        NA
## 839  -1.051247452 1.905048e+00           0.38383838         8        NA
## 840  -1.173238772 2.101467e+00           0.39393939         8        NA
## 841  -1.284168586 2.264946e+00           0.40404040         8        NA
## 842  -1.382533058 2.396493e+00           0.41414141         8        NA
## 843  -1.467619171 2.499091e+00           0.42424242         8        NA
## 844  -1.539376232 2.576694e+00           0.43434343         8        NA
## 845  -1.598263968 2.633416e+00           0.44444444         8        NA
## 846  -1.645097638 2.672971e+00           0.45454545         8        NA
## 847  -1.680907782 2.698332e+00           0.46464646         8        NA
## 848  -1.706828510 2.711568e+00           0.47474747         8        NA
## 849  -1.724024293 2.713781e+00           0.48484848         8        NA
## 850  -1.733661294 2.705088e+00           0.49494949         8        NA
## 851  -1.736925593 2.684636e+00           0.50505051         8        NA
## 852  -1.735087029 2.650620e+00           0.51515152         8        NA
## 853  -1.729603618 2.600352e+00           0.52525253         8        NA
## 854  -1.722257303 2.530435e+00           0.53535354         8        NA
## 855  -1.715307013 2.437074e+00           0.54545455         8        NA
## 856  -1.711639805 2.316603e+00           0.55555556         8        NA
## 857  -1.714895650 2.166215e+00           0.56565657         8        NA
## 858  -1.729537122 1.984850e+00           0.57575758         8        NA
## 859  -1.760833019 1.774086e+00           0.58585859         8        NA
## 860  -1.814726141 1.538823e+00           0.59595960         8        NA
## 861  -1.897561496 1.287500e+00           0.60606061         8        NA
## 862  -2.015662798 1.031638e+00           0.61616162         8        NA
## 863  -2.174762597 7.846285e-01           0.62626263         8        NA
## 864  -2.379313644 5.598883e-01           0.63636364         8        NA
## 865  -2.631734084 3.687400e-01           0.64646465         8        NA
## 866  -2.931663511 2.185123e-01           0.65656566         8        NA
## 867  -3.275327268 1.113807e-01           0.66666667         8        NA
## 868  -3.655120452 4.430399e-02           0.67676768         8        NA
## 869  -4.059539716 1.014012e-02           0.68686869         8        NA
## 870  -4.473757310 3.811234e-08           0.69696970         8        NA
## 871  -4.880634616 4.327794e-03           0.70707071         8        NA
## 872  -5.256492319 6.403609e-03           0.71717172         8        NA
## 873  -5.577896273 3.236816e-03           0.72727273         8        NA
## 874  -5.822608538 5.653030e-08           0.73737374         8        NA
## 875  -5.971199375 7.053188e-08           0.74747475         8        NA
## 876  -6.020682981 0.000000e+00           0.75757576         8        NA
## 877  -5.961018778 1.748579e-08           0.76767677         8        NA
## 878  -5.752827742 2.675722e-03           0.77777778         8        NA
## 879  -5.432183158 3.643918e-03           0.78787879         8        NA
## 880  -5.034473710 3.431355e-04           0.79797980         8        NA
## 881  -4.599117236 5.788151e-03           0.80808081         8        NA
## 882  -4.146323435 3.164594e-02           0.81818182         8        NA
## 883  -3.693973242 8.638670e-02           0.82828283         8        NA
## 884  -3.260910666 1.780004e-01           0.83838384         8        NA
## 885  -2.863061743 3.112032e-01           0.84848485         8        NA
## 886  -2.512271986 4.859754e-01           0.85858586         8        NA
## 887  -2.215884290 6.971486e-01           0.86868687         8        NA
## 888  -1.976838797 9.351001e-01           0.87878788         8        NA
## 889  -1.794168263 1.187332e+00           0.88888889         8        NA
## 890  -1.663762149 1.440492e+00           0.89898990         8        NA
## 891  -1.579270418 1.682329e+00           0.90909091         8        NA
## 892  -1.533027840 1.903178e+00           0.91919192         8        NA
## 893  -1.516901121 2.096743e+00           0.92929293         8        NA
## 894  -1.522990227 2.260170e+00           0.93939394         8        NA
## 895  -1.544146663 2.393573e+00           0.94949495         8        NA
## 896  -1.574300498 2.499237e+00           0.95959596         8        NA
## 897  -1.608611209 2.580744e+00           0.96969697         8        NA
## 898  -1.643473153 2.642204e+00           0.97979798         8        NA
## 899  -1.676414416 2.687665e+00           0.98989899         8        NA
## 900  -1.705928911 2.720766e+00           1.00000000         8        NA
## 901  -1.175292340 1.628360e+00           0.00000000         9        NA
## 902  -1.069900671 1.416706e+00           0.01010101         9        NA
## 903  -0.962625750 1.183620e+00           0.02020202         9        NA
## 904  -0.857022955 9.381958e-01           0.03030303         9        NA
## 905  -0.757000451 6.932082e-01           0.04040404         9        NA
## 906  -0.666561974 4.641730e-01           0.05050505         9        NA
## 907  -0.589504628 2.675623e-01           0.06060606         9        NA
## 908  -0.529106268 1.183676e-01           0.07070707         9        NA
## 909  -0.487866208 2.749204e-02           0.08080808         9        NA
## 910  -0.467602776 2.954616e-04           0.09090909         9        NA
## 911  -0.469107504 3.433133e-02           0.10101010         9        NA
## 912  -0.487888088 1.078653e-01           0.11111111         9        NA
## 913  -0.519096773 1.970101e-01           0.12121212         9        NA
## 914  -0.557264817 2.782283e-01           0.13131313         9        NA
## 915  -0.596473859 3.321333e-01           0.14141414         9        NA
## 916  -0.630777853 3.471010e-01           0.15151515         9        NA
## 917  -0.654687859 3.213057e-01           0.16161616         9        NA
## 918  -0.663651873 2.625087e-01           0.17171717         9        NA
## 919  -0.654477996 1.855464e-01           0.18181818         9        NA
## 920  -0.625656693 1.081141e-01           0.19191919         9        NA
## 921  -0.577545183 4.598784e-02           0.20202020         9        NA
## 922  -0.512381136 9.120503e-03           0.21212121         9        NA
## 923  -0.434014034 7.463827e-04           0.22222222         9        NA
## 924  -0.347874676 1.408148e-02           0.23232323         9        NA
## 925  -0.261360364 2.931556e-02           0.24242424         9        NA
## 926  -0.182108618 3.367059e-02           0.25252525         9        NA
## 927  -0.117624842 2.364453e-02           0.26262626         9        NA
## 928  -0.074852086 6.654700e-03           0.27272727         9        NA
## 929  -0.060498611 1.444676e-03           0.28282828         9        NA
## 930  -0.078615138 3.014676e-02           0.29292929         9        NA
## 931  -0.126811081 1.025411e-01           0.30303030         9        NA
## 932  -0.202445477 2.244393e-01           0.31313131         9        NA
## 933  -0.301733410 3.946546e-01           0.32323232         9        NA
## 934  -0.419882874 6.050558e-01           0.33333333         9        NA
## 935  -0.551468389 8.424150e-01           0.34343434         9        NA
## 936  -0.690855417 1.091136e+00           0.35353535         9        NA
## 937  -0.832604100 1.336032e+00           0.36363636         9        NA
## 938  -0.971808356 1.564502e+00           0.37373737         9        NA
## 939  -1.104343830 1.767770e+00           0.38383838         9        NA
## 940  -1.227013608 1.941165e+00           0.39393939         9        NA
## 941  -1.337593975 2.083632e+00           0.40404040         9        NA
## 942  -1.434793096 2.196818e+00           0.41414141         9        NA
## 943  -1.518142559 2.284017e+00           0.42424242         9        NA
## 944  -1.587845267 2.349227e+00           0.43434343         9        NA
## 945  -1.644603381 2.396418e+00           0.44444444         9        NA
## 946  -1.689447826 2.429064e+00           0.45454545         9        NA
## 947  -1.723587115 2.449889e+00           0.46464646         9        NA
## 948  -1.748288788 2.460759e+00           0.47474747         9        NA
## 949  -1.764802391 2.462660e+00           0.48484848         9        NA
## 950  -1.774328924 2.455705e+00           0.49494949         9        NA
## 951  -1.778038262 2.439134e+00           0.50505051         9        NA
## 952  -1.777132929 2.411329e+00           0.51515152         9        NA
## 953  -1.772953439 2.369834e+00           0.52525253         9        NA
## 954  -1.767116822 2.311467e+00           0.53535354         9        NA
## 955  -1.761675591 2.232562e+00           0.54545455         9        NA
## 956  -1.759279265 2.129401e+00           0.55555556         9        NA
## 957  -1.763315066 1.998880e+00           0.56565657         9        NA
## 958  -1.777999287 1.839344e+00           0.57575758         9        NA
## 959  -1.808387433 1.651504e+00           0.58585859         9        NA
## 960  -1.860271049 1.439191e+00           0.59595960         9        NA
## 961  -1.939933738 1.209720e+00           0.60606061         9        NA
## 962  -2.053749501 9.735939e-01           0.61616162         9        NA
## 963  -2.207623617 7.434506e-01           0.62626263         9        NA
## 964  -2.406299193 5.323179e-01           0.63636364         9        NA
## 965  -2.652579238 3.515076e-01           0.64646465         9        NA
## 966  -2.946541304 2.086487e-01           0.65656566         9        NA
## 967  -3.284845352 1.064060e-01           0.66666667         9        NA
## 968  -3.660252654 4.228608e-02           0.67676768         9        NA
## 969  -4.061491303 9.651665e-03           0.68686869         9        NA
## 970  -4.473757309 0.000000e+00           0.69696970         9        NA
## 971  -4.879674879 4.105398e-03           0.70707071         9        NA
## 972  -5.255461801 6.068569e-03           0.71717172         9        NA
## 973  -5.577355486 3.058357e-03           0.72727273         9        NA
## 974  -5.822608528 3.923457e-08           0.73737374         9        NA
## 975  -5.971199372 1.367182e-08           0.74747475         9        NA
## 976  -6.020682988 1.008301e-07           0.75757576         9        NA
## 977  -5.961018754 5.926804e-08           0.76767677         9        NA
## 978  -5.752962626 2.525507e-03           0.77777778         9        NA
## 979  -5.432309634 3.446207e-03           0.78787879         9        NA
## 980  -5.034503776 3.242338e-04           0.79797980         9        NA
## 981  -4.598956238 5.492810e-03           0.80808081         9        NA
## 982  -4.146601228 3.013427e-02           0.81818182         9        NA
## 983  -3.695943883 8.239109e-02           0.82828283         9        NA
## 984  -3.266108980 1.697910e-01           0.83838384         9        NA
## 985  -2.873041218 2.965301e-01           0.84848485         9        NA
## 986  -2.528368375 4.620882e-01           0.85858586         9        NA
## 987  -2.239027685 6.609287e-01           0.86868687         9        NA
## 988  -2.007434925 8.833150e-01           0.87878788         9        NA
## 989  -1.832058718 1.116977e+00           0.88888889         9        NA
## 990  -1.708260534 1.349154e+00           0.89898990         9        NA
## 991  -1.629260459 1.568486e+00           0.90909091         9        NA
## 992  -1.587102252 1.766372e+00           0.91919192         9        NA
## 993  -1.573517232 1.937584e+00           0.92929293         9        NA
## 994  -1.580619793 2.080197e+00           0.93939394         9        NA
## 995  -1.601402030 2.195000e+00           0.94949495         9        NA
## 996  -1.630025784 2.284661e+00           0.95959596         9        NA
## 997  -1.661934156 2.352865e+00           0.96969697         9        NA
## 998  -1.693819433 2.403593e+00           0.97979798         9        NA
## 999  -1.723490700 2.440626e+00           0.98989899         9        NA
## 1000 -1.749683565 2.467252e+00           1.00000000         9        NA
## 1001 -1.204870218 1.516664e+00           0.00000000        10        NA
## 1002 -1.096837627 1.324085e+00           0.01010101        10        NA
## 1003 -0.986273606 1.110264e+00           0.02020202        10        NA
## 1004 -0.876876215 8.833471e-01           0.03030303        10        NA
## 1005 -0.772758231 6.551411e-01           0.04040404        10        NA
## 1006 -0.678175319 4.403048e-01           0.05050505        10        NA
## 1007 -0.597201481 2.546978e-01           0.06060606        10        NA
## 1008 -0.533388828 1.130383e-01           0.07070707        10        NA
## 1009 -0.489480018 2.632319e-02           0.08080808        10        NA
## 1010 -0.467492146 2.831160e-04           0.09090909        10        NA
## 1011 -0.468338565 3.306292e-02           0.10101010        10        NA
## 1012 -0.487377364 1.042159e-01           0.11111111        10        NA
## 1013 -0.519557636 1.908707e-01           0.12121212        10        NA
## 1014 -0.559153090 2.702080e-01           0.13131313        10        NA
## 1015 -0.599953293 3.232301e-01           0.14141414        10        NA
## 1016 -0.635717370 3.383821e-01           0.15151515        10        NA
## 1017 -0.660692497 3.136558e-01           0.16161616        10        NA
## 1018 -0.670124792 2.564854e-01           0.17171717        10        NA
## 1019 -0.660707474 1.813460e-01           0.18181818        10        NA
## 1020 -0.630919564 1.056201e-01           0.19191919        10        NA
## 1021 -0.581216797 4.485765e-02           0.20202020        10        NA
## 1022 -0.514042637 8.864315e-03           0.21212121        10        NA
## 1023 -0.433563558 7.216025e-04           0.22222222        10        NA
## 1024 -0.345572801 1.368307e-02           0.23232323        10        NA
## 1025 -0.257717963 2.851420e-02           0.24242424        10        NA
## 1026 -0.177917087 3.273196e-02           0.25252525        10        NA
## 1027 -0.113919703 2.294332e-02           0.26262626        10        NA
## 1028 -0.072838627 6.434446e-03           0.27272727        10        NA
## 1029 -0.061467901 1.391755e-03           0.28282828        10        NA
## 1030 -0.083797670 2.909952e-02           0.29292929        10        NA
## 1031 -0.137155107 9.884668e-02           0.30303030        10        NA
## 1032 -0.218560006 2.157769e-01           0.31313131        10        NA
## 1033 -0.323845833 3.780995e-01           0.32323232        10        NA
## 1034 -0.447833421 5.773351e-01           0.33333333        10        NA
## 1035 -0.584743098 8.002954e-01           0.34343434        10        NA
## 1036 -0.728651635 1.031852e+00           0.35353535        10        NA
## 1037 -0.873918557 1.257647e+00           0.36363636        10        NA
## 1038 -1.015535932 1.466132e+00           0.37373737        10        NA
## 1039 -1.149375245 1.649619e+00           0.38383838        10        NA
## 1040 -1.272321947 1.804399e+00           0.39393939        10        NA
## 1041 -1.382302810 1.930143e+00           0.40404040        10        NA
## 1042 -1.478222332 2.028934e+00           0.41414141        10        NA
## 1043 -1.559831400 2.104233e+00           0.42424242        10        NA
## 1044 -1.627554391 2.159989e+00           0.43434343        10        NA
## 1045 -1.682300200 2.199991e+00           0.44444444        10        NA
## 1046 -1.725279522 2.227475e+00           0.45454545        10        NA
## 1047 -1.757846002 2.244931e+00           0.46464646        10        NA
## 1048 -1.781373754 2.254045e+00           0.47474747        10        NA
## 1049 -1.797178985 2.255703e+00           0.48484848        10        NA
## 1050 -1.806489436 2.250011e+00           0.49494949        10        NA
## 1051 -1.810462189 2.236297e+00           0.50505051        10        NA
## 1052 -1.810247755 2.213105e+00           0.51515152        10        NA
## 1053 -1.807095808 2.178198e+00           0.52525253        10        NA
## 1054 -1.802494927 2.128618e+00           0.53535354        10        NA
## 1055 -1.798334785 2.060863e+00           0.54545455        10        NA
## 1056 -1.797074307 1.971256e+00           0.55555556        10        NA
## 1057 -1.801893584 1.856530e+00           0.56565657        10        NA
## 1058 -1.816801651 1.714626e+00           0.57575758        10        NA
## 1059 -1.846667833 1.545592e+00           0.58585859        10        NA
## 1060 -1.897142928 1.352400e+00           0.59595960        10        NA
## 1061 -1.974439752 1.141408e+00           0.60606061        10        NA
## 1062 -2.084952127 9.222125e-01           0.61616162        10        NA
## 1063 -2.234707920 7.067372e-01           0.62626263        10        NA
## 1064 -2.428674994 5.075886e-01           0.63636364        10        NA
## 1065 -2.669966946 3.359846e-01           0.64646465        10        NA
## 1066 -2.959024999 1.997463e-01           0.65656566        10        NA
## 1067 -3.292878929 1.019205e-01           0.66666667        10        NA
## 1068 -3.664610005 4.047460e-02           0.67676768        10        NA
## 1069 -4.063157975 9.216707e-03           0.68686869        10        NA
## 1070 -4.473757309 1.289261e-09           0.69696970        10        NA
## 1071 -4.878849234 3.909059e-03           0.70707071        10        NA
## 1072 -5.254572127 5.773357e-03           0.71717172        10        NA
## 1073 -5.576887026 2.901864e-03           0.72727273        10        NA
## 1074 -5.822608518 0.000000e+00           0.73737374        10        NA
## 1075 -5.971199374 0.000000e+00           0.74747475        10        NA
## 1076 -6.020682964 0.000000e+00           0.75757576        10        NA
## 1077 -5.961018731 0.000000e+00           0.76767677        10        NA
## 1078 -5.753074209 2.394053e-03           0.77777778        10        NA
## 1079 -5.432411660 3.272547e-03           0.78787879        10        NA
## 1080 -5.034527451 3.075745e-04           0.79797980        10        NA
## 1081 -4.598831755 5.231430e-03           0.80808081        10        NA
## 1082 -4.146876974 2.878501e-02           0.81818182        10        NA
## 1083 -3.697695382 7.880516e-02           0.82828283        10        NA
## 1084 -3.270639568 1.624067e-01           0.83838384        10        NA
## 1085 -2.881653082 2.833404e-01           0.84848485        10        NA
## 1086 -2.542158654 4.406823e-01           0.85858586        10        NA
## 1087 -2.258730120 6.286331e-01           0.86868687        10        NA
## 1088 -2.033324751 8.374336e-01           0.87878788        10        NA
## 1089 -1.863927994 1.055095e+00           0.88888889        10        NA
## 1090 -1.745459565 1.269438e+00           0.89898990        10        NA
## 1091 -1.670791338 1.469919e+00           0.90909091        10        NA
## 1092 -1.631744602 1.648856e+00           0.91919192        10        NA
## 1093 -1.619964431 1.801911e+00           0.92929293        10        NA
## 1094 -1.627606025 1.927884e+00           0.93939394        10        NA
## 1095 -1.647805313 2.028057e+00           0.94949495        10        NA
## 1096 -1.674938362 2.105334e+00           0.95959596        10        NA
## 1097 -1.704697940 2.163405e+00           0.96969697        10        NA
## 1098 -1.734029430 2.206089e+00           0.97979798        10        NA
## 1099 -1.760973048 2.236897e+00           0.98989899        10        NA
## 1100 -1.784456553 2.258807e+00           1.00000000        10        NA
## 1101           NA           NA                   NA         0 0.2800000
## 1102           NA           NA                   NA         0 0.8000000
## 1103           NA           NA                   NA         0 0.2200000
## 1104           NA           NA                   NA         0 0.0900000
## 1105           NA           NA                   NA         1 0.2800000
## 1106           NA           NA                   NA         1 0.8000000
## 1107           NA           NA                   NA         1 0.2200000
## 1108           NA           NA                   NA         1 0.0900000
## 1109           NA           NA                   NA         1 0.7474747
## 1110           NA           NA                   NA         2 0.2800000
## 1111           NA           NA                   NA         2 0.8000000
## 1112           NA           NA                   NA         2 0.2200000
## 1113           NA           NA                   NA         2 0.0900000
## 1114           NA           NA                   NA         2 0.7474747
## 1115           NA           NA                   NA         2 0.6969697
## 1116           NA           NA                   NA         3 0.2800000
## 1117           NA           NA                   NA         3 0.8000000
## 1118           NA           NA                   NA         3 0.2200000
## 1119           NA           NA                   NA         3 0.0900000
## 1120           NA           NA                   NA         3 0.7474747
## 1121           NA           NA                   NA         3 0.6969697
## 1122           NA           NA                   NA         3 0.7575758
## 1123           NA           NA                   NA         4 0.2800000
## 1124           NA           NA                   NA         4 0.8000000
## 1125           NA           NA                   NA         4 0.2200000
## 1126           NA           NA                   NA         4 0.0900000
## 1127           NA           NA                   NA         4 0.7474747
## 1128           NA           NA                   NA         4 0.6969697
## 1129           NA           NA                   NA         4 0.7575758
## 1130           NA           NA                   NA         4 0.7676768
## 1131           NA           NA                   NA         5 0.2800000
## 1132           NA           NA                   NA         5 0.8000000
## 1133           NA           NA                   NA         5 0.2200000
## 1134           NA           NA                   NA         5 0.0900000
## 1135           NA           NA                   NA         5 0.7474747
## 1136           NA           NA                   NA         5 0.6969697
## 1137           NA           NA                   NA         5 0.7575758
## 1138           NA           NA                   NA         5 0.7676768
## 1139           NA           NA                   NA         5 0.7575758
## 1140           NA           NA                   NA         6 0.2800000
## 1141           NA           NA                   NA         6 0.8000000
## 1142           NA           NA                   NA         6 0.2200000
## 1143           NA           NA                   NA         6 0.0900000
## 1144           NA           NA                   NA         6 0.7474747
## 1145           NA           NA                   NA         6 0.6969697
## 1146           NA           NA                   NA         6 0.7575758
## 1147           NA           NA                   NA         6 0.7676768
## 1148           NA           NA                   NA         6 0.7575758
## 1149           NA           NA                   NA         6 0.7373737
## 1150           NA           NA                   NA         7 0.2800000
## 1151           NA           NA                   NA         7 0.8000000
## 1152           NA           NA                   NA         7 0.2200000
## 1153           NA           NA                   NA         7 0.0900000
## 1154           NA           NA                   NA         7 0.7474747
## 1155           NA           NA                   NA         7 0.6969697
## 1156           NA           NA                   NA         7 0.7575758
## 1157           NA           NA                   NA         7 0.7676768
## 1158           NA           NA                   NA         7 0.7575758
## 1159           NA           NA                   NA         7 0.7373737
## 1160           NA           NA                   NA         7 0.7474747
## 1161           NA           NA                   NA         8 0.2800000
## 1162           NA           NA                   NA         8 0.8000000
## 1163           NA           NA                   NA         8 0.2200000
## 1164           NA           NA                   NA         8 0.0900000
## 1165           NA           NA                   NA         8 0.7474747
## 1166           NA           NA                   NA         8 0.6969697
## 1167           NA           NA                   NA         8 0.7575758
## 1168           NA           NA                   NA         8 0.7676768
## 1169           NA           NA                   NA         8 0.7575758
## 1170           NA           NA                   NA         8 0.7373737
## 1171           NA           NA                   NA         8 0.7474747
## 1172           NA           NA                   NA         8 0.7474747
## 1173           NA           NA                   NA         9 0.2800000
## 1174           NA           NA                   NA         9 0.8000000
## 1175           NA           NA                   NA         9 0.2200000
## 1176           NA           NA                   NA         9 0.0900000
## 1177           NA           NA                   NA         9 0.7474747
## 1178           NA           NA                   NA         9 0.6969697
## 1179           NA           NA                   NA         9 0.7575758
## 1180           NA           NA                   NA         9 0.7676768
## 1181           NA           NA                   NA         9 0.7575758
## 1182           NA           NA                   NA         9 0.7373737
## 1183           NA           NA                   NA         9 0.7474747
## 1184           NA           NA                   NA         9 0.7474747
## 1185           NA           NA                   NA         9 0.7474747
## 1186           NA           NA                   NA        10 0.2800000
## 1187           NA           NA                   NA        10 0.8000000
## 1188           NA           NA                   NA        10 0.2200000
## 1189           NA           NA                   NA        10 0.0900000
## 1190           NA           NA                   NA        10 0.7474747
## 1191           NA           NA                   NA        10 0.6969697
## 1192           NA           NA                   NA        10 0.7575758
## 1193           NA           NA                   NA        10 0.7676768
## 1194           NA           NA                   NA        10 0.7575758
## 1195           NA           NA                   NA        10 0.7373737
## 1196           NA           NA                   NA        10 0.7474747
## 1197           NA           NA                   NA        10 0.7474747
## 1198           NA           NA                   NA        10 0.7474747
## 1199           NA           NA                   NA        10 0.7575758
##              y_y
## 1             NA
## 2             NA
## 3             NA
## 4             NA
## 5             NA
## 6             NA
## 7             NA
## 8             NA
## 9             NA
## 10            NA
## 11            NA
## 12            NA
## 13            NA
## 14            NA
## 15            NA
## 16            NA
## 17            NA
## 18            NA
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## 20            NA
## 21            NA
## 22            NA
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## 707           NA
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## 710           NA
## 711           NA
## 712           NA
## 713           NA
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## 715           NA
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## 718           NA
## 719           NA
## 720           NA
## 721           NA
## 722           NA
## 723           NA
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```

Reference

Most of the codes are from this website!!!

Reference: https://bearloga.github.io/bayesopt-tutorial-r/